🧠:<think>Okay, so I have this problem about a filmmaker analyzing the spread of three urban legends using differential equations. The system given is:[ begin{cases}frac{dx}{dt} = -a x + b y z frac{dy}{dt} = c x - d y + e z frac{dz}{dt} = -f x y + g zend{cases}]And I need to find the equilibrium points and analyze their stability. Hmm, let me start with Sub-problem 1: finding the equilibrium points.Equilibrium points occur where all the derivatives are zero. So, I need to solve the system:1. (-a x + b y z = 0)2. (c x - d y + e z = 0)3. (-f x y + g z = 0)Alright, so I have three equations with three variables: x, y, z. Let me see how I can solve this.First, equation 1: (-a x + b y z = 0). Let's rewrite that as (a x = b y z), so (x = frac{b}{a} y z). I'll keep that in mind.Equation 3: (-f x y + g z = 0). Let's rewrite that as (f x y = g z), so (z = frac{f}{g} x y). Hmm, interesting.Now, let's substitute equation 1 into equation 3. Since x is expressed in terms of y and z, and z is expressed in terms of x and y, maybe we can find a relationship between y and z.Wait, let me substitute x from equation 1 into equation 3.From equation 1: (x = frac{b}{a} y z). Plugging this into equation 3:(z = frac{f}{g} left( frac{b}{a} y z right) y = frac{f b}{a g} y^2 z)So, (z = frac{f b}{a g} y^2 z). Let's rearrange this:(z - frac{f b}{a g} y^2 z = 0)Factor out z:(z left(1 - frac{f b}{a g} y^2 right) = 0)So, either z = 0 or (1 - frac{f b}{a g} y^2 = 0).Case 1: z = 0.If z = 0, let's see what happens to the other equations.From equation 1: (a x = b y z = 0), so x = 0.From equation 2: (c x - d y + e z = 0). Since x = 0 and z = 0, this simplifies to (-d y = 0), so y = 0.So, one equilibrium point is (0, 0, 0). That makes sense; if all influences are zero, it's an equilibrium.Case 2: (1 - frac{f b}{a g} y^2 = 0).So, (frac{f b}{a g} y^2 = 1), which implies (y^2 = frac{a g}{f b}), so (y = pm sqrt{frac{a g}{f b}}).But since y represents an influence score, which I assume is non-negative, we can take only the positive root: (y = sqrt{frac{a g}{f b}}).Now, let's find x and z in terms of y.From equation 1: (x = frac{b}{a} y z).From equation 3: (z = frac{f}{g} x y).Let me substitute x from equation 1 into equation 3:(z = frac{f}{g} left( frac{b}{a} y z right) y = frac{f b}{a g} y^2 z)Wait, that's the same equation as before, which led us to the condition on y. So, since we already have y, let's compute z.From equation 3: (z = frac{f}{g} x y). But from equation 1, x = (b/a) y z. So, substituting x into equation 3:(z = frac{f}{g} cdot frac{b}{a} y z cdot y = frac{f b}{a g} y^2 z)But we know that (y^2 = frac{a g}{f b}), so:(z = frac{f b}{a g} cdot frac{a g}{f b} z = z)Which is just an identity, so it doesn't give new information. So, we need another equation to relate x and z.Wait, let's use equation 2: (c x - d y + e z = 0).We have y, so let's plug in y = sqrt(a g / (f b)).So, equation 2 becomes:(c x - d sqrt{frac{a g}{f b}} + e z = 0)But from equation 1: (x = frac{b}{a} y z = frac{b}{a} sqrt{frac{a g}{f b}} z)Simplify x:(x = frac{b}{a} cdot sqrt{frac{a g}{f b}} z = frac{b}{a} cdot sqrt{frac{a g}{f b}} z)Let me compute the square root term:(sqrt{frac{a g}{f b}} = sqrt{frac{a}{f} cdot frac{g}{b}})So, x becomes:(x = frac{b}{a} cdot sqrt{frac{a}{f} cdot frac{g}{b}} z = frac{b}{a} cdot sqrt{frac{a g}{f b}} z)Let me square both sides to see if I can find a relationship, but maybe it's easier to express x in terms of z and substitute into equation 2.So, x = (b/a) * sqrt(a g / (f b)) * z.Let me compute (b/a) * sqrt(a g / (f b)).Let me write it as:( frac{b}{a} cdot sqrt{frac{a g}{f b}} = frac{b}{a} cdot sqrt{frac{a}{f} cdot frac{g}{b}} = frac{b}{a} cdot sqrt{frac{a g}{f b}} )Hmm, maybe it's better to compute the numerical factor.Let me denote (k = sqrt{frac{a g}{f b}}), so y = k.Then, x = (b/a) * k * z = (b/a) * sqrt(a g / (f b)) * z.Let me compute (b/a) * sqrt(a g / (f b)):= (b/a) * sqrt(a g / (f b)) = (b/a) * sqrt(a/(f b)) * sqrt(g)= (b/a) * sqrt(a/(f b)) * sqrt(g) = (b/a) * (sqrt(a) / sqrt(f b)) ) * sqrt(g)= (b/a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)Simplify:= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / sqrt(f b)) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)Wait, maybe I can factor out sqrt(b) in the denominator:= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)= (b / a) * (sqrt(a) / (sqrt(f) sqrt(b))) ) * sqrt(g)Hmm, this seems repetitive. Maybe I should compute it numerically.Let me compute the coefficient:Let me denote:C = (b / a) * sqrt(a g / (f b)) = (b / a) * sqrt(a g / (f b)).Let me square C:C^2 = (b^2 / a^2) * (a g / (f b)) = (b^2 / a^2) * (a g / (f b)) = (b^2 * a g) / (a^2 f b) ) = (b g) / (a f)So, C = sqrt( (b g) / (a f) )Therefore, x = C z.So, x = sqrt( (b g) / (a f) ) z.Now, going back to equation 2:c x - d y + e z = 0We have x = C z, y = k = sqrt(a g / (f b)), so:c C z - d k + e z = 0Factor out z:(c C + e) z - d k = 0So,z = (d k) / (c C + e)But k = sqrt(a g / (f b)) and C = sqrt( (b g) / (a f) )Let me compute c C + e:c C = c * sqrt( (b g) / (a f) )So,z = d * sqrt(a g / (f b)) / (c * sqrt( (b g) / (a f) ) + e )Let me simplify the denominator:c * sqrt( (b g) / (a f) ) + e = c * sqrt( (b g) / (a f) ) + eLet me write sqrt( (b g) / (a f) ) as sqrt(b g) / sqrt(a f)So,c * sqrt(b g) / sqrt(a f) + eSo, z = d * sqrt(a g / (f b)) / ( c * sqrt(b g) / sqrt(a f) + e )Let me compute sqrt(a g / (f b)):= sqrt(a g) / sqrt(f b)Similarly, sqrt(b g) / sqrt(a f) = sqrt(b g) / sqrt(a f)So, let me write z as:z = d * (sqrt(a g) / sqrt(f b)) / ( c * (sqrt(b g) / sqrt(a f)) + e )Let me factor out 1 / sqrt(f) from the denominator:Denominator: c * sqrt(b g) / sqrt(a f) + e = (c sqrt(b g) / sqrt(a) ) / sqrt(f) + eSo,z = d * (sqrt(a g) / sqrt(f b)) / ( (c sqrt(b g) / sqrt(a)) / sqrt(f) + e )Let me write this as:z = d * sqrt(a g) / (sqrt(f b)) / ( (c sqrt(b g) / sqrt(a)) / sqrt(f) + e )Let me multiply numerator and denominator by sqrt(f):z = d * sqrt(a g) / (sqrt(f b)) * sqrt(f) / ( c sqrt(b g) / sqrt(a) + e sqrt(f) )Simplify numerator:sqrt(a g) * sqrt(f) / sqrt(f b) = sqrt(a g f) / sqrt(f b) = sqrt(a g f) / (sqrt(f) sqrt(b)) ) = sqrt(a g) / sqrt(b)So, numerator becomes d * sqrt(a g) / sqrt(b)Denominator: c sqrt(b g) / sqrt(a) + e sqrt(f)So,z = (d * sqrt(a g) / sqrt(b)) / ( c sqrt(b g) / sqrt(a) + e sqrt(f) )Let me rationalize the denominator:Multiply numerator and denominator by sqrt(a):z = (d * sqrt(a g) / sqrt(b) * sqrt(a)) / ( c sqrt(b g) + e sqrt(f) sqrt(a) )Simplify numerator:d * sqrt(a g) * sqrt(a) / sqrt(b) = d * a sqrt(g) / sqrt(b)Denominator: c sqrt(b g) + e sqrt(a f)So,z = (d a sqrt(g) / sqrt(b)) / ( c sqrt(b g) + e sqrt(a f) )Let me factor sqrt(g) from the denominator:Denominator: sqrt(g) (c sqrt(b) + e sqrt(a f / g) )Wait, maybe not. Alternatively, let me factor sqrt(b) from the denominator:Denominator: c sqrt(b g) + e sqrt(a f) = sqrt(b) (c sqrt(g) ) + e sqrt(a f)Hmm, not sure if that helps. Maybe leave it as is.So, z is expressed in terms of the constants. Then, x = C z, where C = sqrt( (b g) / (a f) )So, x = sqrt( (b g) / (a f) ) * zSo, x = sqrt( (b g) / (a f) ) * [ (d a sqrt(g) / sqrt(b)) / ( c sqrt(b g) + e sqrt(a f) ) ]Simplify x:= sqrt( (b g) / (a f) ) * (d a sqrt(g) / sqrt(b)) / ( c sqrt(b g) + e sqrt(a f) )= [ sqrt(b g) / sqrt(a f) ) ] * (d a sqrt(g) / sqrt(b)) / ( c sqrt(b g) + e sqrt(a f) )Simplify numerator:sqrt(b g) * d a sqrt(g) / sqrt(b) = d a gDenominator: sqrt(a f) * ( c sqrt(b g) + e sqrt(a f) )So,x = (d a g) / [ sqrt(a f) ( c sqrt(b g) + e sqrt(a f) ) ]Similarly, z was:z = (d a sqrt(g) / sqrt(b)) / ( c sqrt(b g) + e sqrt(a f) )So, x and z are expressed in terms of the constants.Therefore, the non-zero equilibrium point is:x = (d a g) / [ sqrt(a f) ( c sqrt(b g) + e sqrt(a f) ) ]y = sqrt( a g / (f b) )z = (d a sqrt(g) / sqrt(b)) / ( c sqrt(b g) + e sqrt(a f) )Hmm, that's quite complicated. Maybe I can write it more neatly.Let me denote:Let me compute the denominator term:D = c sqrt(b g) + e sqrt(a f)So,x = (d a g) / ( sqrt(a f) D )z = (d a sqrt(g) / sqrt(b)) / DAnd y = sqrt( a g / (f b) )So, the non-zero equilibrium point is:x = (d a g) / ( sqrt(a f) D )y = sqrt( a g / (f b) )z = (d a sqrt(g) / sqrt(b)) / DWhere D = c sqrt(b g) + e sqrt(a f)Alternatively, we can factor out sqrt(g) from D:D = sqrt(g) (c sqrt(b) + e sqrt(a f / g) )But I'm not sure if that helps.Alternatively, let me rationalize the expressions.Wait, maybe I can write x and z in terms of y.Since y = sqrt( a g / (f b) ), let's compute sqrt(a g / (f b)):= sqrt(a g) / sqrt(f b)So, y = sqrt(a g) / sqrt(f b)Then, sqrt(g) = y sqrt(f b) / sqrt(a)Similarly, sqrt(a f) = sqrt(a) sqrt(f)So, let's rewrite x:x = (d a g) / ( sqrt(a f) D )= (d a g) / ( sqrt(a) sqrt(f) (c sqrt(b g) + e sqrt(a f)) )Similarly, z = (d a sqrt(g) / sqrt(b)) / D= (d a sqrt(g)) / ( sqrt(b) (c sqrt(b g) + e sqrt(a f)) )= (d a sqrt(g)) / ( sqrt(b) D )Hmm, maybe not much better.Alternatively, perhaps I can express x and z in terms of y.From equation 1: x = (b/a) y zFrom equation 3: z = (f/g) x ySo, substituting x from equation 1 into equation 3:z = (f/g) * (b/a) y z * y = (f b / (a g)) y^2 zBut we already have y^2 = a g / (f b), so:z = (f b / (a g)) * (a g / (f b)) z = zWhich is consistent, but doesn't help.Alternatively, since y is known, maybe express x and z in terms of y.From equation 1: x = (b/a) y zFrom equation 3: z = (f/g) x ySo, substituting x into equation 3:z = (f/g) * (b/a) y z * y = (f b / (a g)) y^2 zBut y^2 = a g / (f b), so:z = (f b / (a g)) * (a g / (f b)) z = zAgain, just an identity.So, perhaps the only way is to express x and z in terms of the constants as I did before.So, summarizing, the equilibrium points are:1. The trivial equilibrium: (0, 0, 0)2. The non-trivial equilibrium:x = (d a g) / ( sqrt(a f) (c sqrt(b g) + e sqrt(a f)) )y = sqrt( a g / (f b) )z = (d a sqrt(g) / sqrt(b)) / (c sqrt(b g) + e sqrt(a f) )So, that's Sub-problem 1 done.Now, Sub-problem 2: Analyze the stability of each equilibrium point by evaluating the Jacobian matrix.Stability analysis involves finding the Jacobian matrix of the system, evaluating it at each equilibrium point, and then finding the eigenvalues to determine the stability.The Jacobian matrix J is given by the partial derivatives of each equation with respect to x, y, z.So, let's compute J:For the system:dx/dt = -a x + b y zdy/dt = c x - d y + e zdz/dt = -f x y + g zThe Jacobian matrix J is:[ ∂(dx/dt)/∂x , ∂(dx/dt)/∂y , ∂(dx/dt)/∂z ][ ∂(dy/dt)/∂x , ∂(dy/dt)/∂y , ∂(dy/dt)/∂z ][ ∂(dz/dt)/∂x , ∂(dz/dt)/∂y , ∂(dz/dt)/∂z ]So, compute each partial derivative:First row:∂(dx/dt)/∂x = -a∂(dx/dt)/∂y = b z∂(dx/dt)/∂z = b ySecond row:∂(dy/dt)/∂x = c∂(dy/dt)/∂y = -d∂(dy/dt)/∂z = eThird row:∂(dz/dt)/∂x = -f y∂(dz/dt)/∂y = -f x∂(dz/dt)/∂z = gSo, the Jacobian matrix J is:[ -a , b z , b y ][ c , -d , e ][ -f y , -f x , g ]Now, we need to evaluate this matrix at each equilibrium point and find the eigenvalues.First, let's evaluate at the trivial equilibrium (0,0,0):At (0,0,0):J = [ -a , 0 , 0 ] [ c , -d , e ] [ 0 , 0 , g ]So, the Jacobian matrix at (0,0,0) is diagonal except for the c in the (2,1) position and e in the (2,3) position.But more precisely, it's:Row 1: -a, 0, 0Row 2: c, -d, eRow 3: 0, 0, gTo find the eigenvalues, we can look at the diagonal elements if the matrix is diagonal, but it's not. However, since the matrix is block triangular, the eigenvalues are the eigenvalues of the diagonal blocks.Wait, actually, the matrix is:[ -a 0 0 ][ c -d e ][ 0 0 g ]So, it's a block matrix with a 1x1 block (-a) and a 2x2 block:[ -d e ][ 0 g ]So, the eigenvalues are -a, and the eigenvalues of the 2x2 block.The 2x2 block is upper triangular, so its eigenvalues are -d and g.Therefore, the eigenvalues of J at (0,0,0) are -a, -d, and g.Since a, d, g are positive constants, -a and -d are negative, and g is positive.In stability analysis, if all eigenvalues have negative real parts, the equilibrium is stable (attracting). If any eigenvalue has a positive real part, it's unstable.Here, we have eigenvalues: -a (negative), -d (negative), and g (positive). So, since one eigenvalue is positive, the trivial equilibrium (0,0,0) is unstable.Now, let's analyze the non-trivial equilibrium point.Let me denote the non-trivial equilibrium as (x*, y*, z*), where:x* = (d a g) / ( sqrt(a f) D )y* = sqrt( a g / (f b) )z* = (d a sqrt(g) / sqrt(b)) / DWhere D = c sqrt(b g) + e sqrt(a f)So, to evaluate the Jacobian at (x*, y*, z*), we need to compute:J = [ -a , b z* , b y* ] [ c , -d , e ] [ -f y* , -f x* , g ]So, let's plug in the values.First, compute each element:First row:- The (1,1) entry is -a.- The (1,2) entry is b z*.- The (1,3) entry is b y*.Second row:- The (2,1) entry is c.- The (2,2) entry is -d.- The (2,3) entry is e.Third row:- The (3,1) entry is -f y*.- The (3,2) entry is -f x*.- The (3,3) entry is g.So, the Jacobian matrix at (x*, y*, z*) is:[ -a , b z* , b y* ][ c , -d , e ][ -f y* , -f x* , g ]Now, to analyze the stability, we need to find the eigenvalues of this matrix. However, computing eigenvalues for a 3x3 matrix is quite involved, especially with these expressions. Maybe we can look for patterns or simplifications.Alternatively, perhaps we can consider the system's behavior near the equilibrium. But given the complexity, maybe it's better to consider the signs of the trace and determinant, but I'm not sure.Alternatively, perhaps we can look for whether all eigenvalues have negative real parts, which would imply stability.But given the complexity, maybe it's better to note that the Jacobian at the non-trivial equilibrium will have eigenvalues that determine its stability. Since the system is non-linear, the stability depends on the eigenvalues of the Jacobian at that point.Given that the trivial equilibrium is unstable, the non-trivial equilibrium could be stable or unstable depending on the parameters.But without specific values, it's hard to determine the exact stability. However, we can note that if all eigenvalues of the Jacobian at (x*, y*, z*) have negative real parts, then the equilibrium is stable; otherwise, it's unstable.Alternatively, maybe we can look for conditions on the parameters that ensure stability.But perhaps the answer expects us to set up the Jacobian and note that the stability depends on the eigenvalues, which would require solving the characteristic equation.Alternatively, maybe we can consider the trace and determinant.The trace of the Jacobian is the sum of the diagonal elements:Trace = -a - d + gThe determinant of the Jacobian is more complex, but for a 3x3 matrix, it's the sum of the products of the diagonal minus the sum of the products of the off-diagonal.But given the complexity, perhaps it's better to state that the stability depends on the eigenvalues of the Jacobian at (x*, y*, z*), which would require solving the characteristic equation:det(J - λ I) = 0Where I is the identity matrix.So, the characteristic equation is:| -a - λ b z* b y* || c -d - λ e || -f y* -f x* g - λ | = 0Expanding this determinant would give a cubic equation in λ, which is difficult to solve symbolically.Therefore, the stability of the non-trivial equilibrium depends on the roots of this cubic equation, which in turn depend on the parameters a, b, c, d, e, f, g.Given that, we can conclude that:- The trivial equilibrium (0,0,0) is unstable because it has a positive eigenvalue (g).- The non-trivial equilibrium's stability depends on the eigenvalues of its Jacobian, which requires further analysis beyond the scope of this problem without specific parameter values.Alternatively, maybe we can consider the possibility of Hopf bifurcations or other behaviors, but that's probably beyond the current scope.So, summarizing:Sub-problem 1: The equilibrium points are (0,0,0) and a non-trivial point (x*, y*, z*) as derived above.Sub-problem 2: The trivial equilibrium is unstable. The non-trivial equilibrium's stability requires analyzing the eigenvalues of the Jacobian matrix evaluated at (x*, y*, z*), which depends on the specific parameter values.But perhaps the problem expects a more detailed analysis, maybe assuming certain conditions.Alternatively, maybe we can consider the system's behavior in terms of the parameters.Wait, perhaps we can look at the signs of the trace and the determinant.The trace is -a - d + g. If the trace is negative, it's a necessary (but not sufficient) condition for stability.Similarly, the determinant of the Jacobian can give information about the stability.But without specific values, it's hard to conclude.Alternatively, maybe we can consider the system's behavior near the non-trivial equilibrium.But perhaps the answer is that the trivial equilibrium is unstable, and the non-trivial equilibrium's stability depends on the parameters, requiring further analysis.Alternatively, maybe we can consider the possibility that the non-trivial equilibrium is stable if certain conditions on the parameters are met, such as the trace being negative and the determinant being positive, but I'm not sure.Alternatively, perhaps we can note that the non-trivial equilibrium is stable if the real parts of all eigenvalues are negative, which would require that the trace is negative, the determinant is positive, and the sum of the principal minors is positive.But this is getting too involved without specific parameter values.So, perhaps the answer is:- The trivial equilibrium (0,0,0) is unstable.- The non-trivial equilibrium's stability depends on the eigenvalues of the Jacobian matrix evaluated at that point, which requires solving the characteristic equation. Depending on the parameters, it could be stable or unstable.Alternatively, maybe we can look for conditions where the non-trivial equilibrium is stable.But perhaps the answer is that the trivial equilibrium is unstable, and the non-trivial equilibrium is stable if certain conditions on the parameters are satisfied, but without specific values, we can't determine it definitively.Alternatively, maybe we can consider that the non-trivial equilibrium is stable because the influence scores are positive, but that's not necessarily true.Alternatively, perhaps we can note that the non-trivial equilibrium is a saddle point or a stable node or spiral, depending on the parameters.But without more information, it's hard to say.So, perhaps the answer is:- The trivial equilibrium (0,0,0) is unstable.- The non-trivial equilibrium's stability requires evaluating the eigenvalues of the Jacobian matrix at that point, which depends on the specific parameter values. If all eigenvalues have negative real parts, the equilibrium is stable; otherwise, it's unstable.Alternatively, maybe we can consider that the non-trivial equilibrium is stable because the system tends to a steady state where the legends coexist, but that's an assumption.Alternatively, perhaps we can note that the non-trivial equilibrium is stable if the interaction rates are such that the system converges to it, but that's vague.Alternatively, maybe we can compute the characteristic equation and find conditions for stability.But given the time, perhaps it's better to conclude that the trivial equilibrium is unstable, and the non-trivial equilibrium's stability depends on the eigenvalues, which require further analysis.So, in summary:Sub-problem 1: The equilibrium points are (0,0,0) and (x*, y*, z*) as derived.Sub-problem 2: The trivial equilibrium is unstable. The non-trivial equilibrium's stability depends on the eigenvalues of the Jacobian at that point, which requires solving the characteristic equation. Depending on the parameters, it could be stable or unstable.But perhaps the problem expects a more concrete answer, so maybe I should consider that the non-trivial equilibrium is stable if the real parts of all eigenvalues are negative, which would require certain conditions on the parameters.Alternatively, perhaps we can consider that the non-trivial equilibrium is stable because the influence scores are positive and the system reaches a balance, but that's not rigorous.Alternatively, perhaps we can note that the non-trivial equilibrium is a stable spiral or node if the eigenvalues have negative real parts and possibly complex components.But without specific values, it's hard to say.Alternatively, perhaps we can consider that the non-trivial equilibrium is stable if the trace is negative and the determinant is positive, but that's only for 2x2 matrices.Wait, for 3x3 matrices, the Routh-Hurwitz criteria can be used to determine stability based on the coefficients of the characteristic equation.The Routh-Hurwitz criteria state that for a cubic equation:λ^3 + a λ^2 + b λ + c = 0The necessary and sufficient conditions for all roots to have negative real parts are:1. a > 02. b > 03. c > 04. a b > cSo, if we can express the characteristic equation in terms of the coefficients and check these conditions, we can determine stability.So, let's try to compute the characteristic equation for the Jacobian at (x*, y*, z*).The Jacobian is:[ -a - λ , b z* , b y* ][ c , -d - λ , e ][ -f y* , -f x* , g - λ ]The determinant of this matrix is:| -a - λ b z* b y* || c -d - λ e || -f y* -f x* g - λ | = 0Expanding this determinant:Let me denote the matrix as M.The determinant is:(-a - λ) * | (-d - λ)(g - λ) - e*(-f x*) | - b z* * | c (g - λ) - e*(-f y*) | + b y* * | c*(-f x*) - (-d - λ)*(-f y*) |Wait, that's a bit messy. Let me compute each minor.First, expand along the first row.The determinant is:(-a - λ) * M11 - b z* * M12 + b y* * M13Where M11 is the minor for element (1,1):M11 = | (-d - λ) e | | -f x* g - λ |= (-d - λ)(g - λ) - e*(-f x*)= (-d - λ)(g - λ) + e f x*Similarly, M12 is the minor for (1,2):M12 = | c e | | -f y* g - λ |= c (g - λ) - e*(-f y*)= c (g - λ) + e f y*M13 is the minor for (1,3):M13 = | c -d - λ | | -f y* -f x* |= c*(-f x*) - (-d - λ)*(-f y*)= -c f x* - (d + λ) f y*So, putting it all together:det(M) = (-a - λ)[ (-d - λ)(g - λ) + e f x* ] - b z*[ c (g - λ) + e f y* ] + b y*[ -c f x* - (d + λ) f y* ]Let me expand each term:First term: (-a - λ)[ (-d - λ)(g - λ) + e f x* ]= (-a - λ)[ (-d g + d λ - λ g + λ^2) + e f x* ]= (-a - λ)[ λ^2 + (d - g) λ - d g + e f x* ]Second term: - b z*[ c (g - λ) + e f y* ]= -b z* [ c g - c λ + e f y* ]= -b z* c g + b z* c λ - b z* e f y*Third term: b y*[ -c f x* - (d + λ) f y* ]= b y* [ -c f x* - d f y* - f y* λ ]= -b y* c f x* - b y* d f y* - b y* f y* λNow, let's combine all terms:det(M) = (-a - λ)(λ^2 + (d - g) λ - d g + e f x*) - b z* c g + b z* c λ - b z* e f y* - b y* c f x* - b y* d f y* - b y* f y* λLet me expand the first term:= (-a - λ)(λ^2 + (d - g) λ - d g + e f x*)= (-a)(λ^2 + (d - g) λ - d g + e f x*) - λ(λ^2 + (d - g) λ - d g + e f x*)= -a λ^2 - a (d - g) λ + a d g - a e f x* - λ^3 - (d - g) λ^2 + d g λ - e f x* λNow, collect all terms:= -λ^3 - (d - g + a) λ^2 + [ -a (d - g) + d g - e f x* ] λ + (a d g - a e f x*) - b z* c g + b z* c λ - b z* e f y* - b y* c f x* - b y* d f y* - b y* f y* λNow, let's collect like terms:- λ^3 term: -λ^3- λ^2 terms: - (d - g + a) λ^2- λ terms:[ -a (d - g) + d g - e f x* ] λ + b z* c λ - b y* f y* λ= [ -a d + a g + d g - e f x* + b c z* - b f y*^2 ] λ- Constant terms:a d g - a e f x* - b z* c g - b z* e f y* - b y* c f x* - b y* d f y*Now, let's write the characteristic equation as:-λ^3 - (a + d - g) λ^2 + [ -a d + a g + d g - e f x* + b c z* - b f y*^2 ] λ + (a d g - a e f x* - b c g z* - b e f y* z* - b c f x* y* - b d f y*^2 ) = 0Multiply both sides by -1 to make it standard:λ^3 + (a + d - g) λ^2 + [ a d - a g - d g + e f x* - b c z* + b f y*^2 ] λ + (-a d g + a e f x* + b c g z* + b e f y* z* + b c f x* y* + b d f y*^2 ) = 0So, the characteristic equation is:λ^3 + A λ^2 + B λ + C = 0Where:A = a + d - gB = a d - a g - d g + e f x* - b c z* + b f y*^2C = -a d g + a e f x* + b c g z* + b e f y* z* + b c f x* y* + b d f y*^2Now, applying the Routh-Hurwitz criteria for stability:1. A > 02. B > 03. C > 04. A B > CSo, let's check each condition.1. A = a + d - g > 0So, if a + d > g, then A > 0.2. B = a d - a g - d g + e f x* - b c z* + b f y*^2 > 0This is more complex. Let's substitute x*, y*, z*.Recall:x* = (d a g) / ( sqrt(a f) D )y* = sqrt( a g / (f b) )z* = (d a sqrt(g) / sqrt(b)) / DWhere D = c sqrt(b g) + e sqrt(a f)Let me compute each term in B:a d - a g - d g + e f x* - b c z* + b f y*^2Compute term by term:a d - a g - d g = a d - g(a + d)e f x* = e f * (d a g) / ( sqrt(a f) D ) = e f * (d a g) / ( sqrt(a f) D ) = e f * d a g / ( sqrt(a) sqrt(f) D ) = e f * d a g / ( sqrt(a f) D )= e f * d a g / ( sqrt(a f) D ) = e f * d a g / ( sqrt(a f) D ) = e f * d a g / ( sqrt(a f) D )Similarly, -b c z* = -b c * (d a sqrt(g) / sqrt(b)) / D = -b c * d a sqrt(g) / ( sqrt(b) D ) = -c d a sqrt(g) sqrt(b) / DBecause sqrt(b) cancels with 1/sqrt(b) as sqrt(b)/sqrt(b) = 1.Wait, let me compute:z* = (d a sqrt(g) / sqrt(b)) / DSo, -b c z* = -b c * (d a sqrt(g) / sqrt(b)) / D = -c d a sqrt(g) * sqrt(b) / DBecause b / sqrt(b) = sqrt(b)Similarly, b f y*^2 = b f * (a g / (f b)) ) = b f * (a g) / (f b) = a gSo, putting it all together:B = [a d - g(a + d)] + [e f x*] + [ -b c z* ] + [b f y*^2 ]= [a d - g(a + d)] + [ e f x* ] + [ -c d a sqrt(g) sqrt(b) / D ] + [a g ]Simplify:= a d - g a - g d + e f x* - c d a sqrt(b g) / D + a g= a d - g a - g d + a g + e f x* - c d a sqrt(b g) / D= a d - g d + e f x* - c d a sqrt(b g) / DBecause -g a + a g cancels out.So, B = a d - g d + e f x* - c d a sqrt(b g) / DNow, let's compute e f x*:e f x* = e f * (d a g) / ( sqrt(a f) D ) = e f * d a g / ( sqrt(a f) D ) = e f * d a g / ( sqrt(a) sqrt(f) D ) = e f * d a g / ( sqrt(a f) D )= e f * d a g / ( sqrt(a f) D ) = e f * d a g / ( sqrt(a f) D )Similarly, c d a sqrt(b g) / D is as is.So, B = a d - g d + [ e f d a g / ( sqrt(a f) D ) ] - [ c d a sqrt(b g) / D ]Factor out d a / D:= a d - g d + d a / D [ e f g / sqrt(a f) - c sqrt(b g) ]Simplify inside the brackets:e f g / sqrt(a f) = e f g / (sqrt(a) sqrt(f)) ) = e g sqrt(f) / sqrt(a)Similarly, c sqrt(b g) = c sqrt(b) sqrt(g)So,= a d - g d + d a / D [ e g sqrt(f) / sqrt(a) - c sqrt(b) sqrt(g) ]= a d - g d + d a / D [ e g sqrt(f/a) - c sqrt(b g) ]Hmm, not sure if that helps.Alternatively, let me factor sqrt(g):= a d - g d + d a / D [ e g sqrt(f/a) - c sqrt(b) sqrt(g) ]= a d - g d + d a sqrt(g) / D [ e sqrt(f/a) - c sqrt(b) ]Because sqrt(g) is factored out.So,= a d - g d + d a sqrt(g) / D [ e sqrt(f/a) - c sqrt(b) ]Now, let's compute e sqrt(f/a):= e sqrt(f/a) = e sqrt(f) / sqrt(a)Similarly, c sqrt(b) is as is.So,= a d - g d + d a sqrt(g) / D [ e sqrt(f) / sqrt(a) - c sqrt(b) ]= a d - g d + d a sqrt(g) / D [ (e sqrt(f) - c sqrt(a b)) / sqrt(a) ]= a d - g d + d a sqrt(g) / (D sqrt(a)) [ e sqrt(f) - c sqrt(a b) ]= a d - g d + d sqrt(a g) / D [ e sqrt(f) - c sqrt(a b) ]Hmm, this is getting too involved. Maybe it's better to leave B as is.Similarly, C is even more complex.Given the complexity, perhaps it's better to conclude that the non-trivial equilibrium's stability depends on the parameters and requires checking the Routh-Hurwitz conditions, which involve verifying A > 0, B > 0, C > 0, and A B > C.Therefore, summarizing:Sub-problem 1: The equilibrium points are (0,0,0) and (x*, y*, z*) as derived.Sub-problem 2: The trivial equilibrium is unstable. The non-trivial equilibrium's stability requires checking the Routh-Hurwitz conditions based on the parameters.Alternatively, perhaps the non-trivial equilibrium is stable if a + d > g, and other conditions hold, but without specific values, it's hard to say.But perhaps the answer expects us to state that the trivial equilibrium is unstable, and the non-trivial equilibrium's stability depends on the eigenvalues, which requires further analysis.So, in conclusion:Sub-problem 1: The equilibrium points are (0,0,0) and (x*, y*, z*) where x*, y*, z* are as derived.Sub-problem 2: The trivial equilibrium is unstable. The non-trivial equilibrium's stability depends on the eigenvalues of the Jacobian matrix evaluated at that point, which requires solving the characteristic equation. Depending on the parameters, it could be stable or unstable.</think>

🧠:<think>Alright, so I have this problem about optimizing a supply chain network. It's a minimum-cost maximum flow problem, which I remember is a common type of optimization problem in operations research. Let me try to break it down step by step.First, the network is represented as a directed graph G = (V, E). The nodes V are factories, distribution centers, and retail outlets. The edges E have capacities c(u, v), which is the maximum goods that can be transported from u to v. The goal is to maximize the flow from a source node s (a factory) to a sink node t (a retail outlet) while minimizing the cost. Okay, so it's about sending as much flow as possible from s to t with the least cost.Part 1 asks me to formulate this as a minimum-cost maximum flow problem. I need to define variables, constraints, and the objective function using linear programming. Hmm, I think I remember that in linear programming, we define variables for the flow on each edge. So, let's say x_uv is the flow from node u to node v. Each x_uv has to be less than or equal to the capacity c(u, v). Also, we need to make sure that the flow is conserved at each node except the source and sink. That is, for each node v (not s or t), the sum of flows into v equals the sum of flows out of v.The objective function is to minimize the total cost. Each edge has a cost per unit flow, let's call it cost(u, v). So, the total cost is the sum over all edges of cost(u, v) multiplied by x_uv. But wait, we also want to maximize the flow from s to t. So, how do we combine these two objectives? I think in the minimum-cost maximum flow problem, we first maximize the flow, and then among all maximum flows, we choose the one with the minimum cost. Alternatively, it can be formulated as a linear program where we maximize the flow while keeping the cost as low as possible.Wait, actually, in linear programming, we can't have two objectives, so we need to combine them. Maybe we can set the flow as a variable and maximize it, while keeping the cost within a certain limit, but that might not be straightforward. Alternatively, perhaps we can use a two-phase approach: first find the maximum flow, then find the minimum cost for that flow. But I think the standard way is to formulate it as a linear program where we maximize the flow from s to t, subject to flow conservation, capacity constraints, and then also include the cost in the objective function. Hmm, maybe I need to look up the exact formulation.But since I can't look things up, I'll try to recall. The standard minimum-cost flow problem has the objective of minimizing the total cost, subject to flow conservation and capacity constraints. But to make it a maximum flow problem, we can set a lower bound on the flow, but that might complicate things. Alternatively, perhaps we can set the flow to be as large as possible, but I'm not sure how to model that in LP.Wait, maybe I can think of it as a linear program where the objective is to maximize the flow f, subject to the flow conservation constraints and capacity constraints, and also include the cost as part of the constraints. But that doesn't seem right. Alternatively, perhaps I need to use a dual variable approach. The dual variables in linear programming correspond to the shadow prices, which can tell us the sensitivity of the solution to changes in the constraints, like capacities.So, for part 1, I need to define variables x_uv for each edge, constraints that sum of x_uv into a node equals sum of x_vu out of it (except for s and t), and x_uv <= c(u, v). The objective is to maximize the flow from s to t, which is the sum of x_su for all u connected to s, minus the sum of x_us for all u connected to s (but since s is the source, it only sends flow out, so it's just the sum of x_su). But also, we want to minimize the cost, which is sum over all edges of cost(u, v)*x_uv.Hmm, so how do we combine these two objectives? Maybe we can set up the problem as a linear program with two objectives, but that's not standard. Instead, perhaps we can use a weighted sum or prioritize one objective over the other. But I think the standard approach is to first maximize the flow, and then minimize the cost for that maximum flow. Alternatively, we can use a parametric approach where we trade off between cost and flow.Wait, maybe I'm overcomplicating. I think the minimum-cost maximum flow problem can be formulated as a linear program where we maximize the flow f, subject to flow conservation, capacity constraints, and the flow f is the amount leaving the source. So, the variables are x_uv and f. The constraints are:For each node v ≠ s, t:sum_{u} x_uv - sum_{w} x_vw = 0For node s:sum_{v} x_sv = fFor node t:sum_{u} x_ut = fAnd for each edge (u, v):x_uv <= c(u, v)And x_uv >= 0The objective is to maximize f.But wait, that's just the maximum flow problem. To include the cost, we need to minimize the total cost. So, perhaps we need to set up the problem as minimizing the total cost, subject to the flow conservation and capacity constraints, and also ensuring that the flow f is as large as possible. But how?Alternatively, I think the minimum-cost maximum flow problem can be formulated as a linear program where we maximize f, and then minimize the cost. But in LP, we can't have two objectives. So, perhaps we can use a two-phase approach: first find the maximum flow, then find the minimum cost for that flow. But in terms of a single LP, I think we can include both objectives by using a dual variable.Wait, maybe not. Let me think again. The standard minimum-cost flow problem is to find the flow that satisfies certain demands at nodes with minimum cost. In our case, the demand is at the sink t, and the supply is at the source s. So, the problem can be formulated as:Minimize sum_{(u,v) in E} cost(u,v) * x_uvSubject to:For each node v:sum_{(u,v) in E} x_uv - sum_{(v,w) in E} x_vw = b_vWhere b_v is the supply/demand at node v. For the source s, b_s = f (the total flow we want to send), and for the sink t, b_t = -f. For all other nodes, b_v = 0.And for each edge (u, v):x_uv <= c(u, v)x_uv >= 0But in our case, we don't know f in advance. So, perhaps we can include f as a variable and maximize it, while minimizing the cost. But again, that's two objectives. So, maybe we can use a dual variable approach where we consider the dual problem to understand the sensitivity.Wait, perhaps the dual variables correspond to the potential at each node, which can help in finding the shortest paths for augmenting flows. But I'm not sure how to incorporate that into the LP formulation.Alternatively, maybe I can use the fact that in the dual problem, the dual variables correspond to the shadow prices, which can tell us how much the objective function would change if we relax a constraint, like increasing the capacity of an edge. So, for part 1, I need to define the primal LP, and then discuss the dual variables and their interpretation.So, to summarize, the primal LP would have variables x_uv for each edge, representing the flow. The constraints are flow conservation at each node, capacity constraints, and non-negativity. The objective is to minimize the total cost.Wait, but the problem also wants to maximize the flow. So, perhaps the correct approach is to first find the maximum flow, and then find the minimum cost for that flow. Alternatively, we can include a constraint that the flow f is as large as possible, but I'm not sure how to model that.Wait, maybe I can set up the problem as a linear program where we maximize f, the flow from s to t, subject to the flow conservation and capacity constraints, and then also include the cost as part of the objective. But since we can't have two objectives, perhaps we can use a weighted sum, but that's not standard.Alternatively, perhaps the minimum-cost maximum flow problem is a combination of both objectives, where we first maximize the flow and then minimize the cost. So, in terms of linear programming, it's a two-phase problem. But I think in practice, it's formulated as a single LP where we include both objectives with a trade-off parameter. But I'm not sure.Wait, maybe I'm overcomplicating. Let me try to write the primal LP.Primal LP:Variables:x_uv for each edge (u, v) in Ef: the total flow from s to tObjective:Minimize sum_{(u,v) in E} cost(u,v) * x_uvSubject to:For each node v ≠ s, t:sum_{(u,v) in E} x_uv - sum_{(v,w) in E} x_vw = 0For node s:sum_{v} x_sv = fFor node t:sum_{u} x_ut = fFor each edge (u, v):x_uv <= c(u, v)x_uv >= 0And we also want to maximize f. But since we can't have two objectives, perhaps we can set f as a variable and include it in the objective function with a very high coefficient to prioritize maximizing f over minimizing cost. But that's a bit hacky.Alternatively, perhaps the correct approach is to first find the maximum flow f_max, and then find the minimum cost for that flow. But in terms of a single LP, I think we can include both objectives by using a dual variable. Wait, no, dual variables are part of the dual problem, not the primal.Wait, maybe I need to consider the dual problem. The dual of the minimum-cost flow problem would have variables corresponding to the nodes, which are the dual variables or potentials. These potentials can help in determining the reduced costs for each edge, which in turn can help in finding the shortest augmenting paths in algorithms like the successive shortest path algorithm.So, in the dual problem, we have variables y_v for each node v, representing the potential at node v. The dual objective is to maximize the total potential at the sink minus the potential at the source, which corresponds to the minimum cost per unit flow. The constraints are that for each edge (u, v), the potential at v minus the potential at u is less than or equal to the cost of the edge. This ensures that the reduced cost (cost(u,v) - (y_v - y_u)) is non-negative, which is necessary for optimality.So, in the dual problem, we have:Maximize y_t - y_sSubject to:For each edge (u, v):y_v - y_u <= cost(u, v)And y_v can be any real number.The dual variables y_v give us the shadow prices, which indicate how much the total cost would increase if we were to increase the demand at the sink by one unit. This is useful for sensitivity analysis, as it tells us the marginal cost of increasing the flow.So, putting it all together, for part 1, the primal problem is a minimum-cost flow problem where we minimize the total cost subject to flow conservation and capacity constraints, and the dual problem gives us the potentials which can be used to understand the sensitivity of the solution to changes in the network, such as increased demand.Now, moving on to part 2. The demand at the retail outlet (sink t) increases by 20%. So, the new demand is 1.2 times the original demand. We need to analyze how this affects the optimal flow and cost, and suggest a strategy to adjust network capacities.First, increasing the demand means that the sink now requires more flow. In the original problem, the maximum flow was f_max. Now, the new demand is 1.2*f_max. If the network can't handle this increased demand, we'll have a deficit. So, the company needs to either increase capacities on certain edges or add new edges to accommodate the increased flow.But how does this affect the optimal flow and cost? Well, the optimal flow will increase to meet the new demand, but the cost will also increase because we might have to use more expensive edges or increase flow on existing edges with higher costs. The exact impact depends on the network structure and the available capacities.To analyze this, we can use the dual variables from the original problem. The dual variable y_t - y_s gives the marginal cost of increasing the flow by one unit. So, if the demand increases by 20%, the total additional cost would be approximately 0.2*(y_t - y_s)*original_demand. But I'm not sure if it's that straightforward because the dual variables are for the marginal cost, and if the network can't handle the increased demand without increasing capacities, the marginal cost might not be linear.Alternatively, we can perform a sensitivity analysis. If the increased demand is within the current network's capacity, the optimal flow will increase, and the cost will increase by the marginal cost times the increase in demand. But if the network is already at capacity, we'll need to augment the network.So, a practical strategy would be:1. Check if the current network can handle the increased demand by finding the maximum flow with the new demand. If it can, then the optimal flow increases, and the cost increases based on the marginal costs.2. If the current network can't handle the increased demand, identify the bottleneck edges where the flow is already at capacity. These are the edges that are saturated and can't carry more flow.3. To handle the increased demand, the company can either increase the capacities of these bottleneck edges or add new edges to create alternative paths. The choice depends on which option is more cost-effective.4. Another strategy is to adjust the flow on existing edges to find a new optimal flow that meets the increased demand at the minimum cost. This might involve using more expensive edges if necessary.5. Additionally, the company can perform a cost-benefit analysis to determine which edges to upgrade or which new edges to add to minimize the total cost while meeting the increased demand.In terms of the dual variables, the shadow price (marginal cost) tells us how much the total cost will increase for each additional unit of flow. So, if the shadow price is high, it might be more beneficial to invest in increasing capacities rather than paying the high marginal cost.So, to sum up, the increased demand will require either increasing the flow on existing edges (if possible) or augmenting the network by increasing capacities or adding new edges. The dual variables help in understanding the marginal costs and identifying where to invest for the most cost-effective solution.</think>

🧠:<think>Okay, so I have this problem about food contamination levels, and I need to solve two sub-problems. Let me take it step by step.First, Sub-problem 1: I need to find the probability that a randomly selected food sample has a contamination level between 4 ppm and 7 ppm. The contamination levels are normally distributed with a mean (μ) of 5 ppm and a standard deviation (σ) of 1.5 ppm.Hmm, normal distribution. I remember that to find probabilities in a normal distribution, I can use the Z-score formula. The Z-score tells me how many standard deviations an element is from the mean. The formula is Z = (X - μ)/σ.So, for 4 ppm, the Z-score would be (4 - 5)/1.5. Let me calculate that: (4 - 5) is -1, divided by 1.5 is approximately -0.6667. I can write that as -2/3 or approximately -0.67.For 7 ppm, the Z-score would be (7 - 5)/1.5. That's 2 divided by 1.5, which is approximately 1.3333. So that's about 1.33.Now, I need to find the probability that Z is between -0.67 and 1.33. To do this, I can use the standard normal distribution table or a calculator that gives the area under the curve.I think the standard normal table gives the cumulative probability from the left up to a certain Z-score. So, I need to find P(Z ≤ 1.33) and P(Z ≤ -0.67), then subtract the latter from the former to get the probability between them.Looking up Z = 1.33 in the table. Let me recall, the table has Z-scores on the sides and the probabilities inside. For 1.33, I think the value is around 0.9082. Let me double-check: 1.3 gives 0.9032, and 1.33 is a bit higher, so yes, approximately 0.9082.For Z = -0.67, since it's negative, it's on the left side. The table might not have negative Z-scores, but I remember that P(Z ≤ -a) = 1 - P(Z ≤ a). So, first, find P(Z ≤ 0.67). Looking up 0.67, that's about 0.7486. Therefore, P(Z ≤ -0.67) is 1 - 0.7486 = 0.2514.Now, subtracting the two probabilities: 0.9082 - 0.2514 = 0.6568. So, approximately 65.68% probability.Wait, let me make sure I didn't make a mistake. The Z-scores are correct: 4 ppm is (4-5)/1.5 = -0.6667, and 7 ppm is (7-5)/1.5 = 1.3333. The probabilities from the table: 1.33 is indeed around 0.9082, and -0.67 is 1 - 0.7486 = 0.2514. Subtracting gives 0.6568, which is about 65.68%. That seems reasonable because the range from 4 to 7 ppm is symmetric around the mean? Wait, no, 4 is 1 ppm below the mean, and 7 is 2 ppm above. So it's not symmetric. But the probability is still a bit over 65%, which seems plausible.Alternatively, maybe I should use a calculator or more precise Z-table values. Let me check more precise values.For Z = 1.33, the exact value is 0.90820. For Z = -0.67, the exact value is 0.25140. So, 0.90820 - 0.25140 = 0.6568, which is 65.68%. So, that's correct.Alternatively, if I use a calculator, maybe I can get more precise decimal places, but for the purposes of this problem, I think 65.68% is acceptable.So, Sub-problem 1 answer is approximately 65.68%.Moving on to Sub-problem 2: The regulator wants no more than 1% of the samples to exceed a specific contamination level L. So, we need to find L such that P(X > L) = 1%, which is 0.01.In terms of Z-scores, this translates to P(Z > z) = 0.01. So, we need to find the Z-score such that the area to the right of it is 0.01. That means the area to the left is 0.99.Looking at the standard normal table, we need to find the Z-score corresponding to 0.99 cumulative probability. From the table, the Z-score for 0.99 is approximately 2.33. Let me confirm: 2.33 gives about 0.9901, which is very close to 0.99. So, z = 2.33.Now, using the Z-score formula, we can find L.Z = (L - μ)/σWe have Z = 2.33, μ = 5, σ = 1.5.So, 2.33 = (L - 5)/1.5Multiply both sides by 1.5: 2.33 * 1.5 = L - 5Calculating 2.33 * 1.5: 2 * 1.5 is 3, 0.33 * 1.5 is approximately 0.495, so total is 3 + 0.495 = 3.495.So, L - 5 = 3.495Therefore, L = 5 + 3.495 = 8.495 ppm.So, approximately 8.495 ppm. Rounding to a reasonable decimal place, maybe 8.5 ppm.But let me check if the Z-score is exactly 2.33. Sometimes, tables might have more precise values. For example, the exact Z-score for 0.99 is approximately 2.326, which is closer to 2.33. So, 2.326 * 1.5 = ?2.326 * 1.5: 2 * 1.5 = 3, 0.326 * 1.5 = 0.489, so total is 3.489. So, L = 5 + 3.489 = 8.489 ppm, which is approximately 8.49 ppm.Depending on the precision required, we can say 8.49 ppm or round to 8.5 ppm.Let me verify this. If L is 8.49 ppm, then the Z-score is (8.49 - 5)/1.5 = 3.49 / 1.5 ≈ 2.3267, which is the Z-score for 0.99 cumulative probability. So, that's correct.Therefore, the maximum contamination level L is approximately 8.49 ppm.Wait, but let me think again. The problem says "no more than 1% of the food samples should exceed a specific contamination level." So, P(X > L) ≤ 0.01. So, we're looking for the 99th percentile, which is L such that P(X ≤ L) = 0.99. So, yes, that's correct.Alternatively, if I use a calculator or more precise Z-table, the exact Z-score for 0.99 is about 2.326, so L = 5 + 2.326 * 1.5 = 5 + 3.489 = 8.489, which is approximately 8.49 ppm.So, to ensure that no more than 1% exceed L, the maximum contamination level should be approximately 8.49 ppm.I think that's correct. Let me just recap:For Sub-problem 1, we calculated the probability between 4 and 7 ppm by converting to Z-scores, looked up the probabilities, and subtracted to get approximately 65.68%.For Sub-problem 2, we needed the value L such that only 1% of samples exceed it. That corresponds to the 99th percentile, which translates to a Z-score of approximately 2.33, leading to L ≈ 8.49 ppm.I think that's solid. I don't see any mistakes in the calculations.</think>

🧠:<think>Okay, so I have two problems to solve here, both given by Alex to help Jamie develop transferrable skills. Let me tackle them one by one.Starting with the first problem about the technology industry and analyzing the time complexity of a recursive algorithm. The algorithm is defined by the recurrence relation:( T(n) = 3Tleft(frac{n}{2}right) + n^2 )Alex suggested using the Master Theorem, which I remember is a tool for solving recurrence relations of the form ( T(n) = aT(n/b) + f(n) ). The theorem helps determine the asymptotic behavior, like Big O notation, which tells us how the algorithm's running time grows with the input size n.So, let me recall the Master Theorem. It states that for a recurrence relation ( T(n) = aT(n/b) + f(n) ), where a ≥ 1, b > 1, and f(n) is an asymptotically positive function, the time complexity can be determined based on how f(n) compares to ( n^{log_b a} ).There are three cases:1. If ( f(n) = O(n^{log_b a - epsilon}) ) for some ε > 0, then ( T(n) = Theta(n^{log_b a}) ).2. If ( f(n) = Theta(n^{log_b a} log^k n) ) for some k ≥ 0, then ( T(n) = Theta(n^{log_b a} log^{k+1} n) ).3. If ( f(n) = Omega(n^{log_b a + epsilon}) ) for some ε > 0, and if a*f(n/b) ≤ c*f(n) for some c < 1 and all sufficiently large n, then ( T(n) = Theta(f(n)) ).So, applying this to our problem. Here, a = 3, b = 2, and f(n) = n².First, compute ( log_b a ), which is ( log_2 3 ). I remember that ( log_2 3 ) is approximately 1.58496.Now, compare f(n) with ( n^{log_b a} ). So, f(n) is n², and ( n^{log_b a} ) is approximately n^1.58496.So, n² is a higher order term than n^1.58496. That means f(n) grows faster than ( n^{log_b a} ).Looking back at the Master Theorem cases, this would fall under case 3. Because f(n) is asymptotically larger than ( n^{log_b a} ). But wait, I need to check the regularity condition as well. The regularity condition is that a*f(n/b) ≤ c*f(n) for some c < 1 and sufficiently large n.Let's compute a*f(n/b):a = 3, f(n/b) = f(n/2) = (n/2)^2 = n²/4.So, a*f(n/b) = 3*(n²/4) = (3/4)n².Compare this to f(n) = n². So, (3/4)n² ≤ c*n². This is true for c = 3/4, which is less than 1. So, the regularity condition is satisfied.Therefore, by case 3 of the Master Theorem, the time complexity T(n) is Θ(f(n)) = Θ(n²).Wait, but let me double-check. Because sometimes when f(n) is polynomially larger, it's case 3, but sometimes people get confused between case 2 and 3. Since n² is polynomially larger than n^1.58496, and the regularity condition holds, case 3 applies, so T(n) is Θ(n²). That seems right.Moving on to the second problem in the renewable energy sector. Jamie is supposed to model the energy output of a solar farm over time with the differential equation:( frac{dE}{dt} = -0.1E + 5sinleft(frac{pi t}{12}right) )with the initial condition E(0) = 10.So, this is a linear first-order differential equation. The standard form is:( frac{dE}{dt} + P(t)E = Q(t) )Let me rewrite the given equation to match this form.Starting with:( frac{dE}{dt} = -0.1E + 5sinleft(frac{pi t}{12}right) )Bring the -0.1E term to the left:( frac{dE}{dt} + 0.1E = 5sinleft(frac{pi t}{12}right) )So, P(t) = 0.1 and Q(t) = 5 sin(π t /12).To solve this, I can use an integrating factor. The integrating factor μ(t) is given by:( mu(t) = e^{int P(t) dt} = e^{int 0.1 dt} = e^{0.1 t} )Multiply both sides of the differential equation by μ(t):( e^{0.1 t} frac{dE}{dt} + 0.1 e^{0.1 t} E = 5 e^{0.1 t} sinleft(frac{pi t}{12}right) )The left side is the derivative of (e^{0.1 t} E) with respect to t. So, we can write:( frac{d}{dt} left( e^{0.1 t} E right) = 5 e^{0.1 t} sinleft(frac{pi t}{12}right) )Now, integrate both sides with respect to t:( e^{0.1 t} E = int 5 e^{0.1 t} sinleft(frac{pi t}{12}right) dt + C )So, I need to compute the integral on the right. Let me denote this integral as I:( I = int 5 e^{0.1 t} sinleft(frac{pi t}{12}right) dt )To solve this integral, I can use integration by parts or look for a formula for integrating products of exponentials and sinusoids. I remember that integrals of the form ( int e^{at} sin(bt) dt ) can be solved using a standard technique, often involving integration by parts twice and then solving for the integral.Let me recall the formula. The integral ( int e^{at} sin(bt) dt ) is:( frac{e^{at}}{a^2 + b^2} (a sin(bt) - b cos(bt)) + C )Similarly, for cosine, it's similar but with a sign change.So, in our case, a = 0.1 and b = π/12.Therefore, the integral I becomes:( 5 times frac{e^{0.1 t}}{(0.1)^2 + (pi/12)^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + C )Let me compute the denominator:(0.1)^2 = 0.01(π/12)^2 ≈ (3.1416/12)^2 ≈ (0.2618)^2 ≈ 0.0685So, denominator ≈ 0.01 + 0.0685 = 0.0785But maybe it's better to keep it exact for now.So, denominator is ( (0.1)^2 + (pi/12)^2 = 0.01 + pi^2 / 144 )So, putting it all together:( I = 5 times frac{e^{0.1 t}}{0.01 + pi^2 / 144} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + C )Simplify the constants:First, 5 divided by (0.01 + π²/144). Let me compute 0.01 as 1/100, and π²/144 is approximately (9.8696)/144 ≈ 0.0686.So, 1/100 + 0.0686 ≈ 0.0786.But let's compute it exactly:0.01 = 1/100, π²/144 is exact, so 1/100 + π²/144. Let me write it as a single fraction:Find a common denominator, which is 100*144=14400.So, 1/100 = 144/14400, π²/144 = 100π² /14400.Thus, denominator is (144 + 100π²)/14400.Therefore, 5 divided by denominator is 5 * (14400)/(144 + 100π²) = (72000)/(144 + 100π²).Hmm, that's a bit messy, but maybe we can leave it as is for now.So, putting it back into the expression for I:( I = frac{72000}{144 + 100pi^2} e^{0.1 t} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + C )But perhaps it's better to factor out the constants:Let me factor out 0.1 and π/12:( I = frac{72000}{144 + 100pi^2} e^{0.1 t} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + C )Alternatively, factor out 0.1:( I = frac{72000}{144 + 100pi^2} e^{0.1 t} times 0.1 left( sinleft(frac{pi t}{12}right) - frac{pi}{1.2} cosleft(frac{pi t}{12}right) right) + C )Wait, 0.1 is 1/10, and π/12 divided by 0.1 is (π/12)/(1/10) = (π/12)*10 = (5π)/6 ≈ 2.618.But maybe it's better to just keep it as it is.So, going back, we have:( e^{0.1 t} E = I + C )So,( e^{0.1 t} E = frac{72000}{144 + 100pi^2} e^{0.1 t} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + C )Divide both sides by e^{0.1 t}:( E(t) = frac{72000}{144 + 100pi^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + C e^{-0.1 t} )Now, apply the initial condition E(0) = 10.At t = 0:( E(0) = frac{72000}{144 + 100pi^2} left( 0.1 sin(0) - frac{pi}{12} cos(0) right) + C e^{0} = 10 )Simplify:sin(0) = 0, cos(0) = 1.So,( E(0) = frac{72000}{144 + 100pi^2} left( 0 - frac{pi}{12} right) + C = 10 )Compute the first term:( frac{72000}{144 + 100pi^2} times left( -frac{pi}{12} right) = -frac{72000 pi}{12 (144 + 100pi^2)} )Simplify 72000 /12 = 6000.So,( -frac{6000 pi}{144 + 100pi^2} )Thus,( -frac{6000 pi}{144 + 100pi^2} + C = 10 )Solve for C:( C = 10 + frac{6000 pi}{144 + 100pi^2} )So, putting it all together, the solution is:( E(t) = frac{72000}{144 + 100pi^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + left( 10 + frac{6000 pi}{144 + 100pi^2} right) e^{-0.1 t} )This is the general solution. However, we can simplify it further by combining the constants.Let me compute the coefficients numerically to make it more understandable.First, compute the denominator: 144 + 100π².π ≈ 3.1416, so π² ≈ 9.8696.100π² ≈ 986.96.So, denominator ≈ 144 + 986.96 ≈ 1130.96.So, 72000 / 1130.96 ≈ let's compute that.72000 / 1130.96 ≈ 72000 / 1131 ≈ approximately 63.66.Similarly, 6000π / 1130.96 ≈ 6000*3.1416 / 1130.96 ≈ 18849.6 / 1130.96 ≈ 16.67.So, let's plug these approximate values back into the equation.First term:( frac{72000}{1130.96} approx 63.66 )So,( 63.66 times (0.1 sin(pi t /12) - (π/12) cos(π t /12)) )Compute 0.1 * 63.66 ≈ 6.366Compute π/12 ≈ 0.2618, so 0.2618 * 63.66 ≈ 16.66So, the first part becomes approximately:6.366 sin(π t /12) - 16.66 cos(π t /12)The second term is:10 + 16.67 ≈ 26.67, multiplied by e^{-0.1 t}So, putting it all together:E(t) ≈ 6.366 sin(π t /12) - 16.66 cos(π t /12) + 26.67 e^{-0.1 t}But let me see if I can write this in a more compact form, perhaps expressing the sinusoidal terms as a single sine or cosine function with a phase shift.The expression A sin(x) + B cos(x) can be written as C sin(x + φ), where C = sqrt(A² + B²) and tanφ = B/A.Wait, actually, it's A sin(x) + B cos(x) = C sin(x + φ), where C = sqrt(A² + B²) and φ = arctan(B/A) or something like that. Let me recall the exact identity.Yes, the identity is:A sin x + B cos x = C sin(x + φ), where C = sqrt(A² + B²) and φ = arctan(B/A) if A ≠ 0.Alternatively, it can also be written as C cos(x - φ'), depending on the convention.But let's compute it for our case.We have:6.366 sin(π t /12) - 16.66 cos(π t /12)Let me denote A = 6.366, B = -16.66.So, C = sqrt(A² + B²) = sqrt(6.366² + 16.66²)Compute 6.366² ≈ 40.5316.66² ≈ 277.56So, C ≈ sqrt(40.53 + 277.56) = sqrt(318.09) ≈ 17.84Then, φ = arctan(B/A) = arctan(-16.66 / 6.366) ≈ arctan(-2.618)Compute arctan(-2.618). Since arctan is an odd function, arctan(-x) = -arctan(x). So, arctan(2.618) ≈ 69 degrees (since tan(60) ≈ 1.732, tan(69) ≈ 2.605, which is close to 2.618). So, approximately 69 degrees, which is about 1.204 radians.Thus, φ ≈ -1.204 radians.Therefore, the expression can be written as:17.84 sin(π t /12 - 1.204)But since sine is periodic and we can adjust the phase, it's also equivalent to:17.84 cos(π t /12 + φ'), but maybe it's better to stick with sine.So, putting it all together, the solution is approximately:E(t) ≈ 17.84 sin(π t /12 - 1.204) + 26.67 e^{-0.1 t}But let me check the exact expression.Alternatively, since the coefficient of cos is negative, it might be better to write it as a cosine with a phase shift.But regardless, the key point is that the solution consists of a transient exponential term and a steady-state sinusoidal term.So, as t increases, the exponential term e^{-0.1 t} decays to zero, and the energy output approaches the sinusoidal function.Therefore, the general solution is:E(t) = [some sinusoidal function] + [transient exponential term]But to write it exactly, we can keep the constants as they are.Alternatively, we can factor out the exponential term.Wait, let me see. The exact solution is:E(t) = frac{72000}{144 + 100pi^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + left( 10 + frac{6000 pi}{144 + 100pi^2} right) e^{-0.1 t}But perhaps we can write it in terms of a single sinusoidal function plus the exponential.Alternatively, we can leave it in the form we derived.But for the purposes of the answer, maybe it's better to present it in the exact form without approximating the constants.So, let me write it again:E(t) = frac{72000}{144 + 100pi^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + left( 10 + frac{6000 pi}{144 + 100pi^2} right) e^{-0.1 t}Alternatively, factor out the common denominator:Let me denote D = 144 + 100π².Then,E(t) = frac{72000}{D} (0.1 sin(π t /12) - (π/12) cos(π t /12)) + (10 + 6000π / D) e^{-0.1 t}We can write 72000/D as 72000/(144 + 100π²), and 6000π/D as 6000π/(144 + 100π²).But perhaps we can simplify 72000/D:72000 = 720 * 100 = 72 * 10 * 100 = 72 * 1000 = 72000.Wait, 72000/D = 72000/(144 + 100π²) = (72000/100)/(1.44 + π²) = 720/(1.44 + π²)Similarly, 6000π/D = 6000π/(144 + 100π²) = (6000/100)π/(1.44 + π²) = 60π/(1.44 + π²)So, E(t) can be written as:E(t) = frac{720}{1.44 + pi^2} (0.1 sin(π t /12) - (π/12) cos(π t /12)) + left( 10 + frac{60pi}{1.44 + pi^2} right) e^{-0.1 t}This might be a slightly cleaner way to present it.Alternatively, factor out 0.1 from the first term:E(t) = frac{720}{1.44 + pi^2} * 0.1 [sin(π t /12) - (π /1.2) cos(π t /12)] + [10 + (60π)/(1.44 + π²)] e^{-0.1 t}But π /1.2 ≈ 2.618, as before.Alternatively, we can write it as:E(t) = frac{72}{1.44 + pi^2} [sin(π t /12) - (π /1.2) cos(π t /12)] + [10 + (60π)/(1.44 + π²)] e^{-0.1 t}Because 720 * 0.1 = 72.So, E(t) = frac{72}{1.44 + pi^2} [sin(π t /12) - (π /1.2) cos(π t /12)] + left(10 + frac{60pi}{1.44 + pi^2}right) e^{-0.1 t}This seems a bit more compact.Alternatively, we can compute the numerical values of the constants to make it more explicit.Compute 1.44 + π² ≈ 1.44 + 9.8696 ≈ 11.3096.So,72 / 11.3096 ≈ 6.36660π / 11.3096 ≈ 60*3.1416 /11.3096 ≈ 188.496 /11.3096 ≈ 16.67So, E(t) ≈ 6.366 [sin(π t /12) - 2.618 cos(π t /12)] + (10 + 16.67) e^{-0.1 t}Which simplifies to:E(t) ≈ 6.366 sin(π t /12) - 16.66 cos(π t /12) + 26.67 e^{-0.1 t}As before.Alternatively, we can write the sinusoidal part as a single sine function with a phase shift, as I did earlier.So, combining the sine and cosine terms:Let me denote:A = 6.366B = -16.66Then, the amplitude C = sqrt(A² + B²) ≈ sqrt(6.366² + 16.66²) ≈ sqrt(40.53 + 277.56) ≈ sqrt(318.09) ≈ 17.84The phase angle φ = arctan(B/A) = arctan(-16.66 /6.366) ≈ arctan(-2.618) ≈ -1.204 radians.So, the sinusoidal part can be written as 17.84 sin(π t /12 - 1.204)Thus, the solution becomes:E(t) ≈ 17.84 sin(π t /12 - 1.204) + 26.67 e^{-0.1 t}This is a more compact form, showing the steady-state oscillation and the decaying exponential.So, putting it all together, the energy output E(t) is approximately:E(t) ≈ 17.84 sin(π t /12 - 1.204) + 26.67 e^{-0.1 t}But to be precise, we should keep the exact form unless told otherwise. So, maybe it's better to present the exact solution without approximating the constants.Alternatively, we can write it as:E(t) = frac{72}{1.44 + pi^2} sinleft(frac{pi t}{12} - phiright) + left(10 + frac{60pi}{1.44 + pi^2}right) e^{-0.1 t}where ( phi = arctanleft(frac{pi /1.2}{1}right) ), but this might complicate things.Alternatively, just leave it in the form with sine and cosine.So, in conclusion, the exact solution is:E(t) = frac{72000}{144 + 100pi^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + left( 10 + frac{6000 pi}{144 + 100pi^2} right) e^{-0.1 t}But for practical purposes, the approximate form with the combined sinusoidal term is more useful.So, summarizing, the time complexity of the algorithm is Θ(n²), and the energy output E(t) is given by the solution to the differential equation, which includes a decaying exponential and a sinusoidal function.Final Answer1. The time complexity is boxed{Theta(n^2)}.2. The energy output function is boxed{E(t) = frac{72000}{144 + 100pi^2} left( 0.1 sinleft(frac{pi t}{12}right) - frac{pi}{12} cosleft(frac{pi t}{12}right) right) + left( 10 + frac{6000 pi}{144 + 100pi^2} right) e^{-0.1 t}}.</think>