🧠:<think>Alright, so I have this problem about Alex's furniture business. He uses a differential equation to model his sales based on community engagement. The equation given is:[ frac{dS}{dt} = k cdot sqrt{C(t)} cdot S(t) ]And ( C(t) ) is a quadratic function: ( C(t) = a t^2 + b t + c ). The initial condition is ( S(0) = S_0 ). I need to find an expression for ( S(t) ) in terms of the given variables.Hmm, okay. So, this is a differential equation where the rate of change of sales is proportional to the square root of the community engagement score multiplied by the current sales. That makes sense because higher engagement would lead to more word-of-mouth referrals, hence more sales.First, let me write down the equation again:[ frac{dS}{dt} = k cdot sqrt{a t^2 + b t + c} cdot S(t) ]This looks like a separable differential equation. I can rewrite it as:[ frac{dS}{S(t)} = k cdot sqrt{a t^2 + b t + c} , dt ]Yes, so I can integrate both sides. The left side with respect to ( S ) and the right side with respect to ( t ).So, integrating both sides:[ int frac{1}{S} , dS = int k cdot sqrt{a t^2 + b t + c} , dt ]The left integral is straightforward. It should be:[ ln |S| + C_1 = int k cdot sqrt{a t^2 + b t + c} , dt ]But since we're dealing with sales, ( S ) is positive, so we can drop the absolute value:[ ln S = int k cdot sqrt{a t^2 + b t + c} , dt + C_1 ]Now, to solve the right integral, which is ( int sqrt{a t^2 + b t + c} , dt ). This seems a bit tricky. I remember that integrals of the form ( sqrt{at^2 + bt + c} ) can be solved using standard techniques, maybe completing the square or using substitution.Let me try completing the square for the quadratic inside the square root.Given ( a t^2 + b t + c ), completing the square:First, factor out ( a ):[ a left( t^2 + frac{b}{a} t right) + c ]Now, to complete the square inside the parentheses:Take half of ( frac{b}{a} ), which is ( frac{b}{2a} ), square it: ( frac{b^2}{4a^2} ).So, add and subtract this inside the parentheses:[ a left( t^2 + frac{b}{a} t + frac{b^2}{4a^2} - frac{b^2}{4a^2} right) + c ]Simplify:[ a left( left( t + frac{b}{2a} right)^2 - frac{b^2}{4a^2} right) + c ]Distribute the ( a ):[ a left( t + frac{b}{2a} right)^2 - frac{b^2}{4a} + c ]So, the quadratic can be written as:[ a left( t + frac{b}{2a} right)^2 + left( c - frac{b^2}{4a} right) ]Let me denote ( D = c - frac{b^2}{4a} ). So, the expression becomes:[ a left( t + frac{b}{2a} right)^2 + D ]Therefore, the integral becomes:[ int sqrt{a left( t + frac{b}{2a} right)^2 + D} , dt ]This is a standard integral form. The integral of ( sqrt{A x^2 + B} ) is known. Let me recall the formula:[ int sqrt{A x^2 + B} , dx = frac{x}{2} sqrt{A x^2 + B} + frac{B}{2 sqrt{A}} ln left( x sqrt{A} + sqrt{A x^2 + B} right) + C ]But in our case, the variable is ( t + frac{b}{2a} ), so let me make a substitution to simplify.Let ( u = t + frac{b}{2a} ). Then, ( du = dt ).So, the integral becomes:[ int sqrt{a u^2 + D} , du ]Which is:[ frac{u}{2} sqrt{a u^2 + D} + frac{D}{2 sqrt{a}} ln left( u sqrt{a} + sqrt{a u^2 + D} right) + C ]Now, substituting back ( u = t + frac{b}{2a} ):[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{a left( t + frac{b}{2a} right)^2 + D} + frac{D}{2 sqrt{a}} ln left( sqrt{a} left( t + frac{b}{2a} right) + sqrt{a left( t + frac{b}{2a} right)^2 + D} right) + C ]Simplify this expression:First, note that ( D = c - frac{b^2}{4a} ), so:[ sqrt{a left( t + frac{b}{2a} right)^2 + D} = sqrt{a t^2 + b t + c} ]Which is the original expression under the square root. So, that term simplifies back to ( sqrt{C(t)} ).Similarly, ( sqrt{a} left( t + frac{b}{2a} right) = sqrt{a} t + frac{b}{2 sqrt{a}} ). Hmm, not sure if that simplifies further, but let's keep it as is for now.So, putting it all together, the integral is:[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) + C ]Therefore, going back to our earlier equation:[ ln S = k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) right] + C_1 ]Now, exponentiating both sides to solve for ( S(t) ):[ S(t) = e^{k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) right] + C_1} ]Simplify the exponent:[ S(t) = e^{C_1} cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) right] right) ]Let me denote ( e^{C_1} ) as a constant, say ( S_0 ), because when ( t = 0 ), ( S(0) = S_0 ). Let's verify that.At ( t = 0 ):[ S(0) = e^{C_1} cdot expleft( k left[ frac{1}{2} left( 0 + frac{b}{2a} right) sqrt{C(0)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} cdot 0 + frac{b}{2 sqrt{a}} + sqrt{C(0)} right) right] right) ]But ( C(0) = a cdot 0 + b cdot 0 + c = c ). So,[ S(0) = e^{C_1} cdot expleft( k left[ frac{b}{4a} sqrt{c} + frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]But we know ( S(0) = S_0 ). Therefore,[ S_0 = e^{C_1} cdot expleft( k left[ frac{b}{4a} sqrt{c} + frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]So, solving for ( e^{C_1} ):[ e^{C_1} = S_0 cdot expleft( -k left[ frac{b}{4a} sqrt{c} + frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]Therefore, plugging back into ( S(t) ):[ S(t) = S_0 cdot expleft( -k left[ frac{b}{4a} sqrt{c} + frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) right] right) ]This simplifies to:[ S(t) = S_0 cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) - frac{b}{4a} sqrt{c} - frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]Hmm, this is getting quite complicated. Maybe there's a better way to express this.Alternatively, perhaps I can write the integral as:[ int sqrt{a t^2 + b t + c} , dt = frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) + C ]So, the integral is:[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) + C ]Therefore, the expression for ( S(t) ) is:[ S(t) = S_0 cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) - frac{1}{2} left( 0 + frac{b}{2a} right) sqrt{C(0)} - frac{D}{2 sqrt{a}} ln left( sqrt{a} cdot 0 + frac{b}{2 sqrt{a}} + sqrt{C(0)} right) right] right) ]Simplifying further, since ( C(0) = c ):[ S(t) = S_0 cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} - frac{b}{4a} sqrt{c} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) - frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]This is the expression for ( S(t) ). It's quite involved, but I think this is the most simplified form unless there's a substitution or further simplification I can do.Alternatively, maybe I can factor out some terms. Let's see:Let me denote ( E = frac{D}{2 sqrt{a}} ), so:[ S(t) = S_0 cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} - frac{b}{4a} sqrt{c} + E ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) - E ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]Which can be written as:[ S(t) = S_0 cdot expleft( frac{k}{2} left( t + frac{b}{2a} right) sqrt{C(t)} - frac{k b}{4a} sqrt{c} + k E ln left( frac{ sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} }{ frac{b}{2 sqrt{a}} + sqrt{c} } right) right) ]Hmm, that might be a slightly cleaner way to present it.But perhaps I should leave it in terms of ( D ) and ( a ), as substituting ( E ) might not necessarily make it clearer.Alternatively, maybe I can write the entire exponent as a single logarithm, but that might not be straightforward.Alternatively, perhaps I can factor out the ( k ) and write it as:[ S(t) = S_0 cdot expleft( k cdot text{[Integral expression evaluated from 0 to t]} right) ]But I think the expression I have is as simplified as it can get without more specific information about ( a ), ( b ), ( c ), etc.So, to recap, the solution to the differential equation is:[ S(t) = S_0 cdot expleft( k left[ frac{1}{2} left( t + frac{b}{2a} right) sqrt{C(t)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} t + frac{b}{2 sqrt{a}} + sqrt{C(t)} right) - frac{1}{2} left( frac{b}{2a} right) sqrt{c} - frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]Where ( D = c - frac{b^2}{4a} ).I think this is the expression for ( S(t) ) in terms of ( t ), ( a ), ( b ), ( c ), ( k ), and ( S_0 ). It might be a bit complex, but it's the result of solving the differential equation with the given quadratic ( C(t) ).Now, moving on to part 2 of the problem. Alex is organizing a community event in June, which will boost ( C(t) ) to 100 for the entire month. The event starts on June 1st, which is ( t = 0 ) for the start of the year. Wait, hold on. If ( t = 0 ) is the start of the year, then June would be ( t = 5 ) months (assuming January is t=0). Wait, no, the problem says "consider ( t = 0 ) as the start of the year", so June would be t=5 if t is in months. But the event is for the entire month of June, so from t=5 to t=6? Or is the event starting on June 1st, which is t=5, and lasting the entire month, so t=5 to t=6. But the problem says "the engagement score before and after the event follows the original quadratic form". So, during June, ( C(t) = 100 ), and before and after, it's the quadratic.Wait, but the problem says "the event starts on June 1st (consider ( t = 0 ) as the start of the year)". Hmm, so if ( t = 0 ) is the start of the year, then June 1st is t=5 (if t is in months). But the problem says "the event starts on June 1st (consider ( t = 0 ) as the start of the year)". So, maybe they are redefining t=0 as June 1st? That would make the event last from t=0 to t=1 (since June has 30 days, but in terms of months, it's one month). Hmm, the problem is a bit ambiguous.Wait, let me read it again:"Assume that Alex wants to increase his sales by organizing a large community event, which he predicts will temporarily boost the community engagement score to ( C(t) = 100 ) for the entire month of June. If the event starts on June 1st (consider ( t = 0 ) as the start of the year) and the engagement score before and after the event follows the original quadratic form, calculate the total increase in sales, ( Delta S ), during the month of June."So, "consider ( t = 0 ) as the start of the year", so June 1st is t=5 (if t is in months). But the event is for the entire month of June, so from t=5 to t=6. However, the problem says "the event starts on June 1st (consider ( t = 0 ) as the start of the year)". So, perhaps they are shifting the time variable so that t=0 corresponds to June 1st. That is, they are redefining t=0 as June 1st, making June the period from t=0 to t=1.But the original differential equation is defined with t as months since the start of the year. So, if we redefine t=0 as June 1st, then the original quadratic ( C(t) ) would have to be adjusted accordingly. But the problem says "the engagement score before and after the event follows the original quadratic form". So, perhaps during June, ( C(t) = 100 ), and before and after, it's the quadratic ( C(t) = a t^2 + b t + c ).But the timing is a bit confusing. Let me parse it again."the event starts on June 1st (consider ( t = 0 ) as the start of the year)".Wait, so June 1st is t=5 if t is in months since the start of the year. But the problem says "consider ( t = 0 ) as the start of the year", so June 1st is t=5. Then, the event is for the entire month of June, so from t=5 to t=6. So, during t=5 to t=6, ( C(t) = 100 ). Before t=5 and after t=6, ( C(t) = a t^2 + b t + c ).So, to calculate the total increase in sales during June, which is from t=5 to t=6, we need to compute ( S(6) - S(5) ).But to compute this, we need to know the expression for ( S(t) ) both before and during the event.Wait, but the expression for ( S(t) ) is given by the solution we found in part 1, which is valid when ( C(t) = a t^2 + b t + c ). However, during June, ( C(t) = 100 ), so the differential equation becomes:[ frac{dS}{dt} = k cdot sqrt{100} cdot S(t) = 10 k S(t) ]Which is a simple exponential growth equation. So, during June, the solution is:[ S(t) = S(5) cdot e^{10 k (t - 5)} ]Therefore, the sales at the end of June (t=6) would be:[ S(6) = S(5) cdot e^{10 k (6 - 5)} = S(5) cdot e^{10 k} ]Therefore, the increase in sales during June is:[ Delta S = S(6) - S(5) = S(5) cdot (e^{10 k} - 1) ]But to find ( Delta S ), we need to know ( S(5) ), which is the sales at the start of June, computed using the original differential equation with ( C(t) = a t^2 + b t + c ).So, first, I need to compute ( S(5) ) using the expression from part 1, then compute ( S(6) ) using the exponential growth during June, and then find the difference.Alternatively, perhaps I can compute the integral of the differential equation from t=5 to t=6 with ( C(t) = 100 ), and then subtract the integral from t=5 to t=6 with ( C(t) = a t^2 + b t + c ). But no, because the differential equation is nonlinear, so the solution isn't simply the integral of the rate.Wait, actually, the differential equation is:[ frac{dS}{dt} = k sqrt{C(t)} S(t) ]Which is a linear differential equation, so the solution is:[ S(t) = S(t_0) cdot expleft( int_{t_0}^t k sqrt{C(tau)} , dtau right) ]Therefore, the increase in sales during June is:[ Delta S = S(6) - S(5) = S(5) cdot left( expleft( int_{5}^{6} k sqrt{C(tau)} , dtau right) - 1 right) ]But during June, ( C(tau) = 100 ), so:[ Delta S = S(5) cdot left( expleft( int_{5}^{6} k cdot 10 , dtau right) - 1 right) = S(5) cdot left( e^{10 k} - 1 right) ]So, to find ( Delta S ), I need ( S(5) ), which is the sales at t=5, computed using the original ( C(t) = a t^2 + b t + c ).Therefore, ( S(5) ) can be found using the expression from part 1:[ S(5) = S_0 cdot expleft( k left[ frac{1}{2} left( 5 + frac{b}{2a} right) sqrt{C(5)} + frac{D}{2 sqrt{a}} ln left( sqrt{a} cdot 5 + frac{b}{2 sqrt{a}} + sqrt{C(5)} right) - frac{1}{2} left( frac{b}{2a} right) sqrt{c} - frac{D}{2 sqrt{a}} ln left( frac{b}{2 sqrt{a}} + sqrt{c} right) right] right) ]Where ( C(5) = a cdot 25 + b cdot 5 + c ).But this expression is quite complicated. Maybe there's a better way to express ( S(5) ).Alternatively, perhaps I can write ( S(t) ) as:[ S(t) = S_0 cdot expleft( k int_{0}^{t} sqrt{C(tau)} , dtau right) ]Wait, is that correct? Let me check.From the differential equation:[ frac{dS}{dt} = k sqrt{C(t)} S(t) ]This is a linear ODE, and its solution is:[ S(t) = S(0) cdot expleft( int_{0}^{t} k sqrt{C(tau)} , dtau right) ]Yes, that's correct. So, actually, my earlier solution was complicating things unnecessarily. The solution is simply:[ S(t) = S_0 cdot expleft( k int_{0}^{t} sqrt{C(tau)} , dtau right) ]Because the differential equation is separable and linear, and the integral of ( sqrt{C(t)} ) is straightforward.Wait, but earlier I tried to compute the integral and got a complicated expression. But actually, since ( C(t) ) is quadratic, the integral ( int sqrt{C(t)} , dt ) can be expressed in terms of logarithms and square roots, as I did earlier. However, the key point is that ( S(t) ) is the initial sales multiplied by the exponential of the integral of ( k sqrt{C(t)} ) from 0 to t.Therefore, for part 2, the total increase in sales during June is:[ Delta S = S(6) - S(5) = S_0 cdot expleft( k int_{0}^{6} sqrt{C(tau)} , dtau right) - S_0 cdot expleft( k int_{0}^{5} sqrt{C(tau)} , dtau right) ]But during June (t=5 to t=6), ( C(t) = 100 ), so:[ int_{0}^{6} sqrt{C(tau)} , dtau = int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau + int_{5}^{6} sqrt{100} , dtau ]Similarly,[ int_{0}^{5} sqrt{C(tau)} , dtau = int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau ]Therefore,[ Delta S = S_0 cdot expleft( k left[ int_{0}^{5} sqrt{C(tau)} , dtau + int_{5}^{6} 10 , dtau right] right) - S_0 cdot expleft( k int_{0}^{5} sqrt{C(tau)} , dtau right) ]Factor out ( S_0 cdot expleft( k int_{0}^{5} sqrt{C(tau)} , dtau right) ):[ Delta S = S_0 cdot expleft( k int_{0}^{5} sqrt{C(tau)} , dtau right) cdot left( expleft( 10 k right) - 1 right) ]But ( S(5) = S_0 cdot expleft( k int_{0}^{5} sqrt{C(tau)} , dtau right) ), so:[ Delta S = S(5) cdot left( e^{10 k} - 1 right) ]Which is the same result as before.Therefore, the total increase in sales during June is ( S(5) cdot (e^{10 k} - 1) ).But to express ( Delta S ) in terms of the given variables, we need to express ( S(5) ) using the expression from part 1.However, since the problem doesn't provide specific values for ( a ), ( b ), ( c ), ( k ), or ( S_0 ), we can only express ( Delta S ) in terms of these variables.Alternatively, if we consider that the original expression for ( S(t) ) is:[ S(t) = S_0 cdot expleft( k int_{0}^{t} sqrt{C(tau)} , dtau right) ]Then, ( S(5) = S_0 cdot expleft( k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau right) )And ( S(6) = S_0 cdot expleft( k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau + k int_{5}^{6} 10 , dtau right) = S(5) cdot e^{10 k} )Therefore, ( Delta S = S(6) - S(5) = S(5) (e^{10 k} - 1) )But since ( S(5) ) is expressed in terms of the integral from 0 to 5, which is a function of ( a ), ( b ), ( c ), ( k ), and ( S_0 ), we can write:[ Delta S = S_0 cdot expleft( k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau right) cdot (e^{10 k} - 1) ]But this is still quite involved. Alternatively, if we denote ( I = int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau ), then:[ Delta S = S_0 cdot e^{k I} cdot (e^{10 k} - 1) ]But without specific values, this is as far as we can go.Alternatively, perhaps the problem expects a different approach. Maybe it assumes that the increase in sales during June is the integral of the differential equation during that month, but that's not accurate because the differential equation is nonlinear.Wait, actually, the differential equation is linear in ( S(t) ), so the solution is multiplicative. Therefore, the increase is indeed ( S(5) (e^{10 k} - 1) ).But to express this in terms of the original variables, we can write:[ Delta S = S_0 cdot expleft( k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau right) cdot (e^{10 k} - 1) ]But this is still complicated. Maybe the problem expects a different interpretation.Wait, another thought: perhaps the total increase in sales is the integral of the derivative over the month of June. That is:[ Delta S = int_{5}^{6} frac{dS}{dt} , dt = int_{5}^{6} k sqrt{C(t)} S(t) , dt ]But since ( C(t) = 100 ) during June, this becomes:[ Delta S = int_{5}^{6} 10 k S(t) , dt ]But ( S(t) ) during June is ( S(5) e^{10 k (t - 5)} ), so:[ Delta S = int_{5}^{6} 10 k S(5) e^{10 k (t - 5)} , dt = 10 k S(5) int_{0}^{1} e^{10 k tau} , dtau ]Let ( tau = t - 5 ), so:[ Delta S = 10 k S(5) cdot left[ frac{e^{10 k tau}}{10 k} right]_0^1 = S(5) (e^{10 k} - 1) ]Which is the same result as before. So, regardless of the approach, the increase in sales is ( S(5) (e^{10 k} - 1) ).Therefore, the total increase in sales during June is:[ Delta S = S(5) (e^{10 k} - 1) ]Where ( S(5) ) is given by:[ S(5) = S_0 cdot expleft( k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau right) ]But since the problem asks for ( Delta S ) in terms of the given variables, and without specific values, this is the expression.However, perhaps the problem expects a numerical answer, but since no specific values are given, I think the answer should be expressed in terms of ( S(5) ), which itself is in terms of the integral of ( sqrt{C(t)} ).Alternatively, maybe the problem assumes that the integral from 0 to 5 can be expressed using the expression from part 1, but that would just bring us back to the same complex expression.Alternatively, perhaps the problem expects us to recognize that the increase is proportional to ( S(5) ), and since ( S(5) ) is based on the original quadratic, the exact form is complicated, but the increase is ( S(5) (e^{10 k} - 1) ).Given that, I think the answer is:[ Delta S = S_0 cdot left( e^{k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau + 10 k} - e^{k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau} right) ]But simplifying, it's:[ Delta S = S_0 cdot e^{k int_{0}^{5} sqrt{a tau^2 + b tau + c} , dtau} cdot (e^{10 k} - 1) ]Which is the same as:[ Delta S = S(5) cdot (e^{10 k} - 1) ]So, in conclusion, the total increase in sales during June is ( S(5) ) multiplied by ( (e^{10 k} - 1) ).But since the problem asks to calculate ( Delta S ), and without specific values, this is the most precise answer we can give.Therefore, the final answer is:[ Delta S = S(5) cdot (e^{10 k} - 1) ]Where ( S(5) ) is computed using the expression from part 1.But perhaps the problem expects a different approach, maybe assuming that the integral from 0 to 5 is negligible or something, but I don't think so.Alternatively, maybe the problem expects us to compute the difference between the sales with the boosted ( C(t) ) and without it. But without specific values, it's impossible to compute numerically.Therefore, I think the answer is as above.</think>

🧠:<think>Alright, so I have this problem about a computer science student analyzing the social impact of cryptocurrencies. They've collected data on transaction volumes and social media sentiment scores. The first part is about finding the Fourier series representation of the transaction volume function, and the second part is solving a differential equation involving the sentiment score. Let me try to work through this step by step.Starting with problem 1: The transaction volume V(t) is given as a periodic function with period T. The function is V(t) = A sin(2πt/T) + B. They want the Fourier series representation over one period T.Hmm, Fourier series. I remember that any periodic function can be represented as a sum of sines and cosines. The general form is:V(t) = a0 + Σ [an cos(nω0 t) + bn sin(nω0 t)]where ω0 = 2π/T, and the coefficients a0, an, bn are calculated using integrals over one period.But wait, in this case, V(t) is already given as a sine function plus a constant. So, is the Fourier series just the same as the given function? Because if it's already expressed in terms of sine and cosine, then maybe it's its own Fourier series.Let me check. The given V(t) is A sin(2πt/T) + B. So, comparing this to the general Fourier series, the constant term a0 is B. The sine term is A sin(2πt/T), which corresponds to the n=1 term in the Fourier series. So, the Fourier series would have a0 = B, b1 = A, and all other an and bn coefficients are zero.Therefore, the Fourier series representation is just V(t) itself because it's already in the correct form. So, I think that's the answer for part 1.Moving on to problem 2: The student hypothesizes that the sentiment score S(t) is influenced by changes in transaction volume. The differential equation given is:dS/dt + α S = β dV/dtWe need to solve this differential equation for S(t), given the Fourier series of V(t) from part 1.First, let's recall that V(t) is A sin(2πt/T) + B, so its derivative dV/dt is (2πA/T) cos(2πt/T). Because the derivative of sin is cos, and the derivative of a constant B is zero.So, substituting dV/dt into the differential equation:dS/dt + α S = β (2πA/T) cos(2πt/T)So, this is a linear first-order ordinary differential equation. The standard form is:dS/dt + P(t) S = Q(t)In this case, P(t) is α (a constant) and Q(t) is β (2πA/T) cos(2πt/T).To solve this, we can use an integrating factor. The integrating factor μ(t) is given by:μ(t) = exp(∫ P(t) dt) = exp(∫ α dt) = e^{α t}Multiplying both sides of the differential equation by μ(t):e^{α t} dS/dt + α e^{α t} S = β (2πA/T) e^{α t} cos(2πt/T)The left side is the derivative of (e^{α t} S) with respect to t. So, we can write:d/dt [e^{α t} S] = β (2πA/T) e^{α t} cos(2πt/T)Now, we need to integrate both sides with respect to t:∫ d/dt [e^{α t} S] dt = ∫ β (2πA/T) e^{α t} cos(2πt/T) dtSo, the left side simplifies to e^{α t} S. The right side is an integral that we need to compute.Let me focus on the integral:∫ e^{α t} cos(2πt/T) dtThis integral can be solved using integration by parts or by using a standard formula. I remember that the integral of e^{at} cos(bt) dt is e^{at} (a cos(bt) + b sin(bt)) / (a² + b²) + C.Let me verify that. Let’s set:Let’s let u = e^{at}, dv = cos(bt) dtThen du = a e^{at} dt, v = (1/b) sin(bt)Integration by parts gives:∫ e^{at} cos(bt) dt = (e^{at} / b) sin(bt) - (a/b) ∫ e^{at} sin(bt) dtNow, let’s do integration by parts again on the integral on the right:Let u = e^{at}, dv = sin(bt) dtThen du = a e^{at} dt, v = -(1/b) cos(bt)So,∫ e^{at} sin(bt) dt = - (e^{at} / b) cos(bt) + (a/b) ∫ e^{at} cos(bt) dtPutting this back into the previous equation:∫ e^{at} cos(bt) dt = (e^{at} / b) sin(bt) - (a/b) [ - (e^{at} / b) cos(bt) + (a/b) ∫ e^{at} cos(bt) dt ]Simplify:= (e^{at} / b) sin(bt) + (a / b²) e^{at} cos(bt) - (a² / b²) ∫ e^{at} cos(bt) dtNow, let’s denote I = ∫ e^{at} cos(bt) dtSo,I = (e^{at} / b) sin(bt) + (a / b²) e^{at} cos(bt) - (a² / b²) IBring the (a² / b²) I term to the left:I + (a² / b²) I = (e^{at} / b) sin(bt) + (a / b²) e^{at} cos(bt)Factor I:I (1 + a² / b²) = e^{at} [ (1/b) sin(bt) + (a / b²) cos(bt) ]Thus,I = e^{at} [ (1/b) sin(bt) + (a / b²) cos(bt) ] / (1 + a² / b² )Multiply numerator and denominator by b²:I = e^{at} [ b sin(bt) + a cos(bt) ] / (b² + a² )So, yes, the integral is:∫ e^{at} cos(bt) dt = e^{at} (a cos(bt) + b sin(bt)) / (a² + b²) + CTherefore, going back to our problem:∫ e^{α t} cos(2πt/T) dt = e^{α t} [ α cos(2πt/T) + (2π/T) sin(2πt/T) ] / (α² + (2π/T)² ) + CSo, putting it back into the equation:e^{α t} S = β (2πA/T) [ e^{α t} ( α cos(2πt/T) + (2π/T) sin(2πt/T) ) / (α² + (2π/T)² ) ] + CDivide both sides by e^{α t}:S(t) = β (2πA/T) [ ( α cos(2πt/T) + (2π/T) sin(2πt/T) ) / (α² + (2π/T)² ) ] + C e^{-α t}So, that's the general solution. Now, we might need to consider initial conditions to find the constant C. However, the problem doesn't specify any initial conditions, so perhaps we can leave it as the general solution.Alternatively, if we assume that as t approaches infinity, the sentiment score S(t) approaches a steady-state solution, which would mean that the transient term C e^{-α t} goes to zero. So, the steady-state solution would be:S(t) = β (2πA/T) [ ( α cos(2πt/T) + (2π/T) sin(2πt/T) ) / (α² + (2π/T)² ) ]But since the problem doesn't specify initial conditions, I think the answer should include the constant of integration. So, the general solution is:S(t) = [ β (2πA/T) / (α² + (2π/T)² ) ] [ α cos(2πt/T) + (2π/T) sin(2πt/T) ] + C e^{-α t}Alternatively, we can write this in terms of amplitude and phase shift. Let me see.Let’s denote:Let’s factor out the amplitude:Let’s compute the amplitude:The coefficient in front of the sine and cosine terms is:[ β (2πA/T) / (α² + (2π/T)² ) ] times [ α cos(...) + (2π/T) sin(...) ]So, the amplitude is:β (2πA/T) / sqrt(α² + (2π/T)^2 )And the phase shift φ is given by tan φ = (2π/T) / αSo, we can write:S(t) = [ β (2πA/T) / sqrt(α² + (2π/T)^2 ) ] cos(2πt/T - φ) + C e^{-α t}But unless the problem specifies initial conditions, we can't determine φ or C. So, perhaps it's better to leave it in the original form.Wait, but let me check the calculations again to make sure I didn't make any mistakes.Starting from:dS/dt + α S = β dV/dtWe found dV/dt = (2πA/T) cos(2πt/T)So, the equation is:dS/dt + α S = β (2πA/T) cos(2πt/T)We used integrating factor e^{α t}, multiplied both sides, integrated, and got:S(t) = [ β (2πA/T) / (α² + (2π/T)^2 ) ] [ α cos(2πt/T) + (2π/T) sin(2πt/T) ] + C e^{-α t}Yes, that seems correct.Alternatively, we can write the homogeneous and particular solutions. The homogeneous solution is C e^{-α t}, and the particular solution is the steady-state part.So, unless more information is given, this is the general solution.Therefore, I think that's the answer for part 2.Wait, but let me think again. Since V(t) is a Fourier series, maybe we can use the Fourier series approach to solve the differential equation. But in this case, since V(t) is already a single sine term, its derivative is a single cosine term, so the differential equation has a forcing function that's a single cosine term. So, the solution method I used is appropriate.Alternatively, if V(t) had a more complex Fourier series with multiple terms, we would have to solve for each term separately and sum them up. But in this case, since V(t) is just a single sine term, the solution is straightforward.So, I think my approach is correct.To recap:1. The Fourier series of V(t) is V(t) itself because it's already expressed as a sum of sine and cosine terms (in this case, just a sine term and a constant).2. The differential equation is solved using the integrating factor method, resulting in a general solution that includes a transient term and a steady-state oscillatory term.I don't see any mistakes in my reasoning, so I think I'm confident with this solution.</think>

🧠:<think>Alright, so I have this problem about a new tax policy and its impact on businesses. There are two parts: the first one is about calculating the tax revenue for a single business with a profit of 600,000, and the second part is about finding the expected total tax revenue from 100 small businesses with profits following a normal distribution. Let me try to tackle each part step by step.Starting with part 1. The tax policy has a progressive tax rate, which means the tax rate increases as the profit increases. The tax function T(x) is defined piecewise:- 10% on profits from 0 up to 100,000,- 15% on profits from 100,000 up to 500,000,- 20% on profits above 500,000.So, for a business with a profit of 600,000, I need to calculate the tax in each bracket and sum them up.First, the profit is 600,000. Let's break it down:1. The first 100,000 is taxed at 10%. So, tax for this portion is 0.10 * 100,000 = 10,000.2. The next portion is from 100,000 to 500,000, which is 400,000. This is taxed at 15%. So, tax here is 0.15 * 400,000 = 60,000.3. The remaining profit is 600,000 - 500,000 = 100,000. This is taxed at 20%. So, tax here is 0.20 * 100,000 = 20,000.Adding these up: 10,000 + 60,000 + 20,000 = 90,000.Wait, let me double-check that. So, 100k at 10% is 10k, 400k at 15% is 60k, and 100k at 20% is 20k. Yep, that adds up to 90,000. So, the overall tax revenue from this business is 90,000.But hold on, the problem also mentions a revenue function R(x) = 1,000,000 ln(1 + x). Hmm, is this relevant for part 1? Let me read the question again.It says, "determine the overall tax revenue generated from a business with a profit of 600,000." So, I think the revenue function R(x) is separate from the tax calculation. Maybe it's just extra information or perhaps for part 2. Since the tax is calculated based on the profit x, which is given as 600,000, I think I can ignore R(x) for part 1. So, my calculation of 90,000 should be correct.Moving on to part 2. Now, we have 100 small businesses, each with an average profit modeled by a normal distribution N(μ, σ²) with μ = 80,000 and σ² = (25,000)^2. So, the mean profit is 80,000, and the standard deviation is 25,000.We need to calculate the expected total tax revenue from these 100 businesses. Since each business is independent, the expected total tax revenue is 100 times the expected tax revenue from a single business.So, first, I need to find E[T(x)] for a single business, where x is the profit, and then multiply by 100.The tax function T(x) is still the same progressive tax:- 10% on x from 0 to 100,000,- 15% on x from 100,000 to 500,000,- 20% on x above 500,000.But since the profits are normally distributed with mean 80,000 and standard deviation 25,000, the profits are mostly concentrated around 80,000, with some spread.Given that the mean is 80,000, which is below 100,000, but the standard deviation is 25,000, so some businesses might have profits above 100,000.So, to compute E[T(x)], we need to integrate T(x) over the normal distribution. Since T(x) is piecewise, we can break the integral into parts.So, E[T(x)] = integral from 0 to 100,000 of 0.1x * f(x) dx + integral from 100,000 to 500,000 of 0.15x * f(x) dx + integral from 500,000 to infinity of 0.2x * f(x) dx.Where f(x) is the probability density function (PDF) of the normal distribution N(80,000, 25,000²).But integrating this directly might be complicated. Maybe we can find the expected value by considering the different tax brackets.Alternatively, we can compute the expected tax by calculating the expected value in each bracket and sum them up.But since the tax is applied progressively, it's not straightforward. Let me think.Wait, actually, the tax is calculated as a function of x, so E[T(x)] = E[ T(x) ] where T(x) is a piecewise function. So, we can express this expectation as:E[T(x)] = E[ T(x) | x < 100,000 ] * P(x < 100,000) + E[ T(x) | 100,000 ≤ x < 500,000 ] * P(100,000 ≤ x < 500,000) + E[ T(x) | x ≥ 500,000 ] * P(x ≥ 500,000)But T(x) is 0.1x for x < 100,000, 0.15x for 100,000 ≤ x < 500,000, and 0.2x for x ≥ 500,000.So, E[T(x)] = E[0.1x | x < 100,000] * P(x < 100,000) + E[0.15x | 100,000 ≤ x < 500,000] * P(100,000 ≤ x < 500,000) + E[0.2x | x ≥ 500,000] * P(x ≥ 500,000)So, we can compute each term separately.First, let's compute P(x < 100,000). Since x ~ N(80,000, 25,000²), we can standardize:Z = (100,000 - 80,000) / 25,000 = 20,000 / 25,000 = 0.8So, P(x < 100,000) = P(Z < 0.8) ≈ 0.7881 (from standard normal table)Similarly, P(100,000 ≤ x < 500,000) = P(x < 500,000) - P(x < 100,000)Compute P(x < 500,000):Z = (500,000 - 80,000) / 25,000 = 420,000 / 25,000 = 16.8That's way beyond the standard normal table, so P(x < 500,000) ≈ 1.Thus, P(100,000 ≤ x < 500,000) ≈ 1 - 0.7881 = 0.2119And P(x ≥ 500,000) ≈ 0, since 500,000 is 16.8 standard deviations above the mean. So, negligible.So, effectively, we can ignore the last term because the probability is almost zero.Therefore, E[T(x)] ≈ E[0.1x | x < 100,000] * 0.7881 + E[0.15x | 100,000 ≤ x < 500,000] * 0.2119Now, we need to compute E[x | x < 100,000] and E[x | 100,000 ≤ x < 500,000]For E[x | x < 100,000], this is the expected value of x truncated at 100,000.Similarly, E[x | 100,000 ≤ x < 500,000] is the expected value of x between 100,000 and 500,000.Calculating these expectations requires integrating x * f(x) over the respective intervals.But since this is a normal distribution, we can use the truncated normal distribution formulas.The expected value of a truncated normal distribution below a point a is:E[x | x < a] = μ - σ * φ((a - μ)/σ) / Φ((a - μ)/σ)Where φ is the standard normal PDF and Φ is the standard normal CDF.Similarly, for the interval between a and b, the expected value is:E[x | a ≤ x < b] = [μ * (Φ((b - μ)/σ) - Φ((a - μ)/σ)) + σ * (φ((a - μ)/σ) - φ((b - μ)/σ))] / (Φ((b - μ)/σ) - Φ((a - μ)/σ))But let me verify that.Wait, actually, for the truncated expectation between a and b, it's:E[x | a ≤ x ≤ b] = [μ * (Φ((b - μ)/σ) - Φ((a - μ)/σ)) + σ * (φ((a - μ)/σ) - φ((b - μ)/σ))] / (Φ((b - μ)/σ) - Φ((a - μ)/σ))Yes, that seems correct.So, let's compute E[x | x < 100,000]:Given μ = 80,000, σ = 25,000, a = 100,000.Compute z = (a - μ)/σ = (100,000 - 80,000)/25,000 = 0.8So, φ(z) = φ(0.8) ≈ 0.2881 (from standard normal table)Φ(z) = Φ(0.8) ≈ 0.7881Thus,E[x | x < 100,000] = μ - σ * φ(z) / Φ(z) = 80,000 - 25,000 * 0.2881 / 0.7881Compute 0.2881 / 0.7881 ≈ 0.3656So, 25,000 * 0.3656 ≈ 9,140Thus, E[x | x < 100,000] ≈ 80,000 - 9,140 ≈ 70,860Wait, that seems a bit low. Let me check the formula again.Wait, actually, the formula is:E[x | x < a] = μ - σ * φ(z) / Φ(z)So, yes, that's correct.So, 80,000 - 25,000*(0.2881 / 0.7881) ≈ 80,000 - 25,000*0.3656 ≈ 80,000 - 9,140 ≈ 70,860. So, that's correct.Now, for E[x | 100,000 ≤ x < 500,000]:Here, a = 100,000, b = 500,000Compute z1 = (a - μ)/σ = 0.8 as beforez2 = (b - μ)/σ = (500,000 - 80,000)/25,000 = 420,000 / 25,000 = 16.8So, φ(z1) = φ(0.8) ≈ 0.2881φ(z2) is φ(16.8), which is practically 0, since the standard normal PDF at 16.8 is negligible.Φ(z1) ≈ 0.7881Φ(z2) ≈ 1So, applying the formula:E[x | a ≤ x ≤ b] = [μ*(Φ(z2) - Φ(z1)) + σ*(φ(z1) - φ(z2))] / (Φ(z2) - Φ(z1))Plugging in the numbers:= [80,000*(1 - 0.7881) + 25,000*(0.2881 - 0)] / (1 - 0.7881)Compute numerator:80,000*(0.2119) = 16,95225,000*(0.2881) = 7,202.5Total numerator ≈ 16,952 + 7,202.5 ≈ 24,154.5Denominator ≈ 0.2119Thus, E[x | 100,000 ≤ x < 500,000] ≈ 24,154.5 / 0.2119 ≈ 113,960Wait, that seems high because the mean is 80,000, but since we're conditioning on x being above 100,000, it's reasonable that the expectation is higher.But let me verify the calculation:Numerator: 80,000*(1 - 0.7881) = 80,000*0.2119 ≈ 16,95225,000*(0.2881 - 0) ≈ 7,202.5Total numerator ≈ 16,952 + 7,202.5 ≈ 24,154.5Denominator ≈ 0.2119So, 24,154.5 / 0.2119 ≈ 113,960. That seems correct.So, now, going back to E[T(x)]:E[T(x)] ≈ 0.1 * E[x | x < 100,000] * P(x < 100,000) + 0.15 * E[x | 100,000 ≤ x < 500,000] * P(100,000 ≤ x < 500,000)Plugging in the numbers:≈ 0.1 * 70,860 * 0.7881 + 0.15 * 113,960 * 0.2119Compute each term:First term: 0.1 * 70,860 = 7,086; then 7,086 * 0.7881 ≈ 7,086 * 0.7881 ≈ let's compute 7,086 * 0.7 = 4,960.2; 7,086 * 0.0881 ≈ 624. So total ≈ 4,960.2 + 624 ≈ 5,584.2Second term: 0.15 * 113,960 ≈ 17,094; then 17,094 * 0.2119 ≈ let's compute 17,094 * 0.2 = 3,418.8; 17,094 * 0.0119 ≈ 203. So total ≈ 3,418.8 + 203 ≈ 3,621.8Adding both terms: 5,584.2 + 3,621.8 ≈ 9,206So, E[T(x)] ≈ 9,206 per business.But wait, let me double-check the calculations:First term:0.1 * 70,860 = 7,0867,086 * 0.7881:Compute 7,086 * 0.7 = 4,960.27,086 * 0.08 = 566.887,086 * 0.0081 ≈ 57.43Adding up: 4,960.2 + 566.88 = 5,527.08 + 57.43 ≈ 5,584.51Second term:0.15 * 113,960 = 17,09417,094 * 0.2119:Compute 17,094 * 0.2 = 3,418.817,094 * 0.0119 ≈ 203.02Adding up: 3,418.8 + 203.02 ≈ 3,621.82Total E[T(x)] ≈ 5,584.51 + 3,621.82 ≈ 9,206.33So, approximately 9,206.33 per business.Therefore, for 100 businesses, the expected total tax revenue is 100 * 9,206.33 ≈ 920,633.Wait, but let me think again. Is this the correct approach?Alternatively, since the tax is a function of x, maybe we can compute E[T(x)] directly by integrating T(x) * f(x) over x from 0 to infinity.But since T(x) is piecewise, we can split the integral into three parts:E[T(x)] = ∫₀^100,000 0.1x * f(x) dx + ∫₁₀₀,₀₀₀^500,000 0.15x * f(x) dx + ∫₅₀₀,₀₀₀^∞ 0.2x * f(x) dxWhich is essentially what I did earlier, breaking it into expectations in each bracket multiplied by their probabilities.But I think my approach is correct.However, let me consider that the tax function is applied progressively, so for x between 100,000 and 500,000, the tax is 0.15x, but actually, it's 0.1*100,000 + 0.15*(x - 100,000). Wait, is that correct?Wait, hold on! I think I made a mistake here. The tax function is defined as:T(x) = 0.1x for x < 100,000T(x) = 0.1*100,000 + 0.15*(x - 100,000) for 100,000 ≤ x < 500,000Similarly, T(x) = 0.1*100,000 + 0.15*(500,000 - 100,000) + 0.2*(x - 500,000) for x ≥ 500,000So, actually, T(x) is not just 0.15x in the second bracket, but rather a flat 10% on the first 100k, plus 15% on the next 400k, and 20% on anything above 500k.Therefore, my initial approach was incorrect because I treated the tax as a flat rate in each bracket, but in reality, it's cumulative.So, for x between 100,000 and 500,000, T(x) = 10,000 + 0.15*(x - 100,000)Similarly, for x ≥ 500,000, T(x) = 10,000 + 60,000 + 0.2*(x - 500,000) = 70,000 + 0.2*(x - 500,000)Therefore, I need to adjust my calculations.So, for part 2, E[T(x)] is:E[T(x)] = E[0.1x | x < 100,000] * P(x < 100,000) + E[10,000 + 0.15(x - 100,000) | 100,000 ≤ x < 500,000] * P(100,000 ≤ x < 500,000) + E[70,000 + 0.2(x - 500,000) | x ≥ 500,000] * P(x ≥ 500,000)So, let's recalculate E[T(x)] with this in mind.First, compute E[0.1x | x < 100,000] * P(x < 100,000):We already have E[x | x < 100,000] ≈ 70,860So, 0.1 * 70,860 ≈ 7,086Multiply by P(x < 100,000) ≈ 0.7881:7,086 * 0.7881 ≈ 5,584.51Second term: E[10,000 + 0.15(x - 100,000) | 100,000 ≤ x < 500,000] * P(100,000 ≤ x < 500,000)First, compute E[10,000 + 0.15(x - 100,000) | 100,000 ≤ x < 500,000]= 10,000 + 0.15 * E[x - 100,000 | 100,000 ≤ x < 500,000]= 10,000 + 0.15 * (E[x | 100,000 ≤ x < 500,000] - 100,000)We already calculated E[x | 100,000 ≤ x < 500,000] ≈ 113,960So, E[x | ... ] - 100,000 ≈ 113,960 - 100,000 = 13,960Thus, 0.15 * 13,960 ≈ 2,094So, the expectation is 10,000 + 2,094 ≈ 12,094Multiply by P(100,000 ≤ x < 500,000) ≈ 0.2119:12,094 * 0.2119 ≈ let's compute 12,000 * 0.2119 = 2,542.8; 94 * 0.2119 ≈ 20. So total ≈ 2,542.8 + 20 ≈ 2,562.8Third term: E[70,000 + 0.2(x - 500,000) | x ≥ 500,000] * P(x ≥ 500,000)But as before, P(x ≥ 500,000) is practically 0, so we can ignore this term.Thus, total E[T(x)] ≈ 5,584.51 + 2,562.8 ≈ 8,147.31So, approximately 8,147.31 per business.Therefore, for 100 businesses, the expected total tax revenue is 100 * 8,147.31 ≈ 814,731.Wait, that's significantly different from my previous calculation. So, I must have made a mistake earlier by not considering the progressive nature of the tax.So, to clarify, in the second bracket, the tax isn't just 0.15x, but rather a base tax of 10,000 plus 15% on the amount over 100,000. Similarly, in the third bracket, it's a base tax plus 20% on the amount over 500,000.Therefore, my initial approach was incorrect because I treated each bracket as a flat rate on the entire x, whereas it's actually a flat rate on the portion within that bracket.So, the correct way is to compute the expected tax as the sum of the base taxes plus the expected taxed amounts in each bracket.Therefore, recalculating:First bracket: x < 100,000, tax is 0.1x. So, E[0.1x | x < 100,000] * P(x < 100,000) ≈ 7,086 * 0.7881 ≈ 5,584.51Second bracket: 100,000 ≤ x < 500,000, tax is 10,000 + 0.15(x - 100,000). So, E[10,000 + 0.15(x - 100,000) | 100,000 ≤ x < 500,000] ≈ 12,094, as calculated earlier. Multiply by P ≈ 0.2119 gives ≈ 2,562.8Third bracket: negligible.Thus, total E[T(x)] ≈ 5,584.51 + 2,562.8 ≈ 8,147.31 per business.Therefore, for 100 businesses, total expected tax revenue ≈ 100 * 8,147.31 ≈ 814,731.But wait, let me verify the calculation for the second term again.E[10,000 + 0.15(x - 100,000) | 100,000 ≤ x < 500,000] = 10,000 + 0.15 * E[x - 100,000 | 100,000 ≤ x < 500,000]We have E[x | 100,000 ≤ x < 500,000] ≈ 113,960Thus, E[x - 100,000 | ... ] = 113,960 - 100,000 = 13,960So, 0.15 * 13,960 ≈ 2,094Thus, total expectation ≈ 10,000 + 2,094 = 12,094Multiply by 0.2119 ≈ 2,562.8Yes, that seems correct.So, total E[T(x)] ≈ 5,584.51 + 2,562.8 ≈ 8,147.31Therefore, 100 businesses would generate approximately 814,731 in tax revenue.But let me think if there's another way to compute this. Maybe using the cumulative distribution function.Alternatively, since the tax is a piecewise linear function, perhaps we can compute the expected tax by integrating T(x) * f(x) over x.But that would involve integrating each piece separately, which is essentially what I did.Alternatively, maybe we can use the fact that E[T(x)] = E[0.1x * I(x < 100,000) + (10,000 + 0.15(x - 100,000)) * I(100,000 ≤ x < 500,000) + (70,000 + 0.2(x - 500,000)) * I(x ≥ 500,000)]Where I(.) is the indicator function.So, E[T(x)] = 0.1 * E[x * I(x < 100,000)] + E[10,000 * I(100,000 ≤ x < 500,000)] + 0.15 * E[(x - 100,000) * I(100,000 ≤ x < 500,000)] + E[70,000 * I(x ≥ 500,000)] + 0.2 * E[(x - 500,000) * I(x ≥ 500,000)]But since P(x ≥ 500,000) is negligible, we can ignore the last two terms.So, E[T(x)] ≈ 0.1 * E[x | x < 100,000] * P(x < 100,000) + 10,000 * P(100,000 ≤ x < 500,000) + 0.15 * E[(x - 100,000) | 100,000 ≤ x < 500,000] * P(100,000 ≤ x < 500,000)Which is exactly what I did earlier.So, plugging in the numbers:0.1 * 70,860 * 0.7881 ≈ 5,584.5110,000 * 0.2119 ≈ 2,1190.15 * 13,960 * 0.2119 ≈ 0.15 * 13,960 ≈ 2,094; 2,094 * 0.2119 ≈ 443. So, total ≈ 2,119 + 443 ≈ 2,562Wait, that doesn't add up. Wait, no, actually, 10,000 * 0.2119 is 2,119, and 0.15 * 13,960 * 0.2119 is approximately 0.15 * 13,960 = 2,094; 2,094 * 0.2119 ≈ 443. So, total ≈ 2,119 + 443 ≈ 2,562Then, adding the first term: 5,584.51 + 2,562 ≈ 8,146.51So, approximately 8,146.51 per business, which aligns with the previous calculation.Therefore, the expected total tax revenue for 100 businesses is approximately 100 * 8,146.51 ≈ 814,651.Rounding to the nearest dollar, it's approximately 814,651.But let me check if I can compute this more accurately.Alternatively, perhaps I can use the fact that the tax function is linear in each bracket and compute the expectations accordingly.But I think my current approach is solid.So, summarizing:For part 1, the tax revenue is 90,000.For part 2, the expected total tax revenue is approximately 814,651.But let me just verify the calculation for E[x | 100,000 ≤ x < 500,000] again.We had:E[x | a ≤ x ≤ b] = [μ*(Φ(z2) - Φ(z1)) + σ*(φ(z1) - φ(z2))] / (Φ(z2) - Φ(z1))With a=100,000, b=500,000, μ=80,000, σ=25,000z1 = (100,000 - 80,000)/25,000 = 0.8z2 = (500,000 - 80,000)/25,000 = 16.8So, Φ(z2) ≈ 1, Φ(z1) ≈ 0.7881φ(z1) ≈ 0.2881, φ(z2) ≈ 0Thus,E[x | a ≤ x ≤ b] = [80,000*(1 - 0.7881) + 25,000*(0.2881 - 0)] / (1 - 0.7881)= [80,000*0.2119 + 25,000*0.2881] / 0.2119= [16,952 + 7,202.5] / 0.2119 ≈ 24,154.5 / 0.2119 ≈ 113,960Yes, that's correct.So, E[x - 100,000 | a ≤ x ≤ b] = 113,960 - 100,000 = 13,960Thus, 0.15 * 13,960 = 2,094So, the second term is 10,000 + 2,094 = 12,094Multiply by 0.2119 ≈ 2,562.8First term: 0.1 * 70,860 ≈ 7,086; 7,086 * 0.7881 ≈ 5,584.51Total E[T(x)] ≈ 5,584.51 + 2,562.8 ≈ 8,147.31So, per business, approximately 8,147.31Thus, 100 businesses: 8,147.31 * 100 = 814,731So, rounding to the nearest dollar, 814,731.But let me consider if the revenue function R(x) = 1,000,000 ln(1 + x) is relevant here. Wait, in part 1, the question is about tax revenue, not the business's revenue. So, R(x) is the business's revenue, but the tax is calculated on the profit x. So, R(x) is separate and not needed for calculating the tax. So, my initial approach was correct.Therefore, the answers are:1. 90,0002. Approximately 814,731But let me check if I can compute the exact value without approximating the integrals.Alternatively, perhaps using the fact that for a normal distribution, the expected value of x truncated at a can be calculated using the formula:E[x | x < a] = μ - σ * φ((a - μ)/σ) / Φ((a - μ)/σ)Similarly, for the interval between a and b, it's:E[x | a ≤ x ≤ b] = [μ * (Φ((b - μ)/σ) - Φ((a - μ)/σ)) + σ * (φ((a - μ)/σ) - φ((b - μ)/σ))] / (Φ((b - μ)/σ) - Φ((a - μ)/σ))So, using these exact formulas, we can compute the expectations more accurately.But since we already used these formulas, our calculations are precise given the standard normal table approximations.Therefore, I think my final answers are correct.</think>

🧠:<think>Okay, so I have this problem about predator-prey models, and I need to modify the prey equation by adding a term that linearly affects the prey population. The original model is given by the system of differential equations:[begin{align*}frac{dx}{dt} &= ax - bxy frac{dy}{dt} &= -cy + dxyend{align*}]where ( x ) is the prey population and ( y ) is the predator population. The constants ( a, b, c, d ) are positive. The first part asks me to incorporate an additional term that linearly affects the prey population. The effect is represented by a constant ( e ) where ( e > 0 ) increases the prey population. So, I need to modify the prey equation.Hmm, so the original prey equation is ( frac{dx}{dt} = ax - bxy ). The term ( ax ) represents the growth rate of the prey, and ( -bxy ) is the predation term. Since ( e ) is supposed to increase the prey population, it should be an additive term. So, I think I need to add ( e ) to the prey equation. That would make the modified prey equation:[frac{dx}{dt} = ax - bxy + e]Is that right? Let me think. If ( e ) is a constant that increases the prey population, it should be a constant term added to the growth rate. So yes, adding ( e ) makes sense. Alternatively, if it's a multiplicative effect, it might be ( a ) increasing, but the problem says it's a linear effect, so additive is more likely.So, the modified system is:[begin{align*}frac{dx}{dt} &= ax - bxy + e frac{dy}{dt} &= -cy + dxyend{align*}]Now, I need to find the new equilibrium points. Equilibrium points occur where both ( frac{dx}{dt} = 0 ) and ( frac{dy}{dt} = 0 ).So, setting ( frac{dx}{dt} = 0 ):[ax - bxy + e = 0]And setting ( frac{dy}{dt} = 0 ):[-cy + dxy = 0]Let me solve these equations simultaneously.First, from the predator equation ( -cy + dxy = 0 ), we can factor out ( y ):[y(-c + dx) = 0]So, either ( y = 0 ) or ( -c + dx = 0 ).Case 1: ( y = 0 )If ( y = 0 ), plug into the prey equation:[ax + e = 0]But ( a ), ( x ), and ( e ) are all positive constants. So, ( ax + e = 0 ) implies ( x = -e/a ), which is negative. Since population can't be negative, this solution is not feasible. So, the only possibility is Case 2.Case 2: ( -c + dx = 0 )Solving for ( x ):[dx = c implies x = frac{c}{d}]So, ( x = c/d ). Now, plug this into the prey equation to find ( y ):[aleft(frac{c}{d}right) - bleft(frac{c}{d}right)y + e = 0]Let me compute each term:First term: ( a(c/d) )Second term: ( -b(c/d)y )Third term: ( e )So, the equation becomes:[frac{ac}{d} - frac{bc}{d} y + e = 0]Let me solve for ( y ):Bring the terms without ( y ) to the other side:[- frac{bc}{d} y = -frac{ac}{d} - e]Multiply both sides by ( -d/(bc) ):[y = frac{ac + ed}{bc}]Simplify:[y = frac{ac}{bc} + frac{ed}{bc} = frac{a}{b} + frac{e}{b}]So, ( y = frac{a + e/d}{b} ). Wait, let me check my algebra.Wait, ( ed/(bc) ) is ( e/(b) times d/c ). Hmm, no, actually, ( ed/(bc) ) is ( e/(b) times (d/c) ). Wait, maybe I made a miscalculation.Wait, starting again:From:[- frac{bc}{d} y = -frac{ac}{d} - e]Multiply both sides by ( -d/(bc) ):Left side: ( y )Right side: ( (ac/d + e) times (d/(bc)) )So,[y = left( frac{ac}{d} + e right) times frac{d}{bc} = frac{ac}{d} times frac{d}{bc} + e times frac{d}{bc} = frac{ac}{bc} + frac{ed}{bc} = frac{a}{b} + frac{ed}{bc}]Yes, that's correct. So,[y = frac{a}{b} + frac{ed}{bc} = frac{a}{b} + frac{e}{b} times frac{d}{c}]Alternatively, factor ( frac{a}{b} ) and ( frac{e}{b} ):[y = frac{a + frac{ed}{c}}{b}]But regardless, the equilibrium point is ( x = c/d ) and ( y = frac{ac + ed}{bc} ).So, the equilibrium points are:1. ( (x, y) = left( frac{c}{d}, frac{ac + ed}{bc} right) )Wait, but earlier when ( y = 0 ), we had an infeasible solution. So, is that the only equilibrium point?Wait, let me think. In the original Lotka-Volterra model, there are two equilibrium points: the trivial one where both populations are zero (which isn't feasible here because ( x ) would be negative if ( y = 0 )) and the non-trivial one where both populations are positive. So, in this modified model, we only have one feasible equilibrium point where both ( x ) and ( y ) are positive.So, that's the only equilibrium point.Wait, but let me confirm. If ( e ) is positive, then ( y ) is positive as well because all constants are positive. So, yes, that's the only feasible equilibrium.So, to recap, the modified system has one equilibrium point at ( x = c/d ) and ( y = (ac + ed)/(bc) ).Moving on to part 2: Determine the stability of the equilibrium points found in sub-problem 1 by analyzing the Jacobian matrix of the modified system.Okay, so to analyze stability, I need to find the Jacobian matrix of the system evaluated at the equilibrium point.First, let's write down the Jacobian matrix. The Jacobian matrix ( J ) is given by:[J = begin{bmatrix}frac{partial}{partial x} left( frac{dx}{dt} right) & frac{partial}{partial y} left( frac{dx}{dt} right) frac{partial}{partial x} left( frac{dy}{dt} right) & frac{partial}{partial y} left( frac{dy}{dt} right)end{bmatrix}]Compute each partial derivative.From the modified prey equation ( frac{dx}{dt} = ax - bxy + e ):- ( frac{partial}{partial x} left( frac{dx}{dt} right) = a - by )- ( frac{partial}{partial y} left( frac{dx}{dt} right) = -bx )From the predator equation ( frac{dy}{dt} = -cy + dxy ):- ( frac{partial}{partial x} left( frac{dy}{dt} right) = dy )- ( frac{partial}{partial y} left( frac{dy}{dt} right) = -c + dx )So, the Jacobian matrix is:[J = begin{bmatrix}a - by & -bx dy & -c + dxend{bmatrix}]Now, evaluate this Jacobian at the equilibrium point ( (x, y) = left( frac{c}{d}, frac{ac + ed}{bc} right) ).Let me compute each entry step by step.First, compute ( a - by ):At equilibrium, ( y = frac{ac + ed}{bc} ). So,[a - b times frac{ac + ed}{bc} = a - frac{b(ac + ed)}{bc} = a - frac{ac + ed}{c}]Simplify:[a - frac{ac}{c} - frac{ed}{c} = a - a - frac{ed}{c} = - frac{ed}{c}]So, the (1,1) entry is ( -ed/c ).Next, compute ( -bx ):At equilibrium, ( x = c/d ). So,[-b times frac{c}{d} = - frac{bc}{d}]So, the (1,2) entry is ( -bc/d ).Now, compute ( dy ):At equilibrium, ( y = frac{ac + ed}{bc} ). So,[d times frac{ac + ed}{bc} = frac{d(ac + ed)}{bc} = frac{ac d + ed^2}{bc} = frac{ac d}{bc} + frac{ed^2}{bc} = frac{a d}{b} + frac{ed}{b}]Wait, let me compute it step by step:[dy = d times frac{ac + ed}{bc} = frac{d(ac + ed)}{bc} = frac{a c d + e d^2}{b c}]Factor out ( d ):[= frac{d(a c + e d)}{b c}]Alternatively, we can leave it as ( frac{a c d + e d^2}{b c} ). Either way, it's a positive value since all constants are positive.So, the (2,1) entry is ( frac{a c d + e d^2}{b c} ).Finally, compute ( -c + dx ):At equilibrium, ( x = c/d ). So,[- c + d times frac{c}{d} = -c + c = 0]So, the (2,2) entry is 0.Putting it all together, the Jacobian matrix at the equilibrium point is:[J = begin{bmatrix}- frac{ed}{c} & - frac{bc}{d} frac{a c d + e d^2}{b c} & 0end{bmatrix}]Simplify the entries:- The (1,1) entry is ( -ed/c )- The (1,2) entry is ( -bc/d )- The (2,1) entry can be simplified:[frac{a c d + e d^2}{b c} = frac{d(a c + e d)}{b c} = frac{d}{b} left( frac{a c + e d}{c} right) = frac{d}{b} left( a + frac{e d}{c} right)]But maybe it's fine as it is.So, the Jacobian is:[J = begin{bmatrix}- frac{ed}{c} & - frac{bc}{d} frac{a c d + e d^2}{b c} & 0end{bmatrix}]To determine the stability, we need to find the eigenvalues of this matrix. The eigenvalues ( lambda ) satisfy the characteristic equation:[det(J - lambda I) = 0]Compute the determinant:[det begin{bmatrix}- frac{ed}{c} - lambda & - frac{bc}{d} frac{a c d + e d^2}{b c} & - lambdaend{bmatrix} = 0]The determinant is:[left( - frac{ed}{c} - lambda right)(- lambda) - left( - frac{bc}{d} times frac{a c d + e d^2}{b c} right) = 0]Simplify each term:First term: ( left( - frac{ed}{c} - lambda right)(- lambda) = lambda left( frac{ed}{c} + lambda right) )Second term: ( - left( - frac{bc}{d} times frac{a c d + e d^2}{b c} right) = frac{bc}{d} times frac{a c d + e d^2}{b c} )Simplify the second term:The ( bc ) cancels with ( b c ), and ( d ) in the denominator cancels with ( d ) in the numerator:[frac{bc}{d} times frac{a c d + e d^2}{b c} = frac{a c d + e d^2}{d} = a c + e d]So, the characteristic equation becomes:[lambda left( frac{ed}{c} + lambda right) - (a c + e d) = 0]Expanding:[lambda cdot frac{ed}{c} + lambda^2 - a c - e d = 0]Rearranged:[lambda^2 + frac{ed}{c} lambda - (a c + e d) = 0]This is a quadratic equation in ( lambda ). Let me write it as:[lambda^2 + left( frac{ed}{c} right) lambda - (a c + e d) = 0]To find the roots, use the quadratic formula:[lambda = frac{ - frac{ed}{c} pm sqrt{ left( frac{ed}{c} right)^2 + 4(a c + e d) } }{2}]Simplify the discriminant:[D = left( frac{ed}{c} right)^2 + 4(a c + e d) = frac{e^2 d^2}{c^2} + 4 a c + 4 e d]Since all constants ( a, b, c, d, e ) are positive, the discriminant ( D ) is positive. Therefore, we have two real eigenvalues.Compute the eigenvalues:[lambda = frac{ - frac{ed}{c} pm sqrt{ frac{e^2 d^2}{c^2} + 4 a c + 4 e d } }{2}]Let me factor out ( frac{1}{c^2} ) inside the square root:[sqrt{ frac{e^2 d^2 + 4 a c^3 + 4 e d c^2 }{c^2} } = frac{ sqrt{ e^2 d^2 + 4 a c^3 + 4 e d c^2 } }{c }]So, the eigenvalues become:[lambda = frac{ - frac{ed}{c} pm frac{ sqrt{ e^2 d^2 + 4 a c^3 + 4 e d c^2 } }{c } }{2 } = frac{ - ed pm sqrt{ e^2 d^2 + 4 a c^3 + 4 e d c^2 } }{ 2 c }]Let me factor out ( c^2 ) from the square root:[sqrt{ e^2 d^2 + 4 a c^3 + 4 e d c^2 } = sqrt{ c^2 (4 a c + 4 e d) + e^2 d^2 } = sqrt{ c^2 (4 a c + 4 e d) + e^2 d^2 }]Hmm, not sure if that helps. Alternatively, let's denote ( A = e d ) and ( B = 2 c sqrt{a c + e d} ), but maybe that's complicating.Alternatively, let me compute the two eigenvalues:First, the positive root:[lambda_1 = frac{ - ed + sqrt{ e^2 d^2 + 4 a c^3 + 4 e d c^2 } }{ 2 c }]Second, the negative root:[lambda_2 = frac{ - ed - sqrt{ e^2 d^2 + 4 a c^3 + 4 e d c^2 } }{ 2 c }]Since the discriminant is positive, and the square root term is larger than ( ed ), the first eigenvalue ( lambda_1 ) will be positive because the numerator is positive (since ( sqrt{...} > ed )), and the denominator is positive. The second eigenvalue ( lambda_2 ) will be negative because both terms in the numerator are negative.So, we have one positive eigenvalue and one negative eigenvalue. In the context of equilibrium points, this means the equilibrium is a saddle point, which is unstable.Wait, but let me think again. In predator-prey models, typically the equilibrium is a center or a spiral depending on the eigenvalues. But in this case, since we have a Jacobian with one positive and one negative eigenvalue, it's a saddle point, which is unstable.But wait, in the original Lotka-Volterra model, the equilibrium is a center, which is neutrally stable, but here, with the addition of the constant term ( e ), the equilibrium becomes a saddle point, which is unstable.Alternatively, maybe I made a mistake in the Jacobian or the eigenvalues.Wait, let's double-check the Jacobian.Original Jacobian:[J = begin{bmatrix}a - by & -bx dy & -c + dxend{bmatrix}]At equilibrium:( x = c/d ), ( y = (ac + ed)/(bc) )So,First entry: ( a - b y = a - b*(ac + ed)/(bc) = a - (ac + ed)/c = a - a - ed/c = -ed/c ) Correct.Second entry: ( -b x = -b*(c/d) = -bc/d ) Correct.Third entry: ( d y = d*(ac + ed)/(bc) = (a c d + e d^2)/(b c) ) Correct.Fourth entry: ( -c + d x = -c + d*(c/d) = -c + c = 0 ) Correct.So, Jacobian is correct.Then, the characteristic equation is:[lambda^2 + (ed/c) lambda - (a c + e d) = 0]Which has two real roots, one positive and one negative.Therefore, the equilibrium point is a saddle point, which is unstable.But wait, in the original Lotka-Volterra model without the constant term, the Jacobian at equilibrium is:[J = begin{bmatrix}0 & -b x^* d y^* & 0end{bmatrix}]Which has eigenvalues ( pm i sqrt{b c} ), purely imaginary, leading to a center (neutral stability). But here, adding the constant term ( e ) changes the Jacobian such that the trace is non-zero, leading to real eigenvalues with opposite signs, hence a saddle point.Therefore, the equilibrium is unstable.But wait, let me think about the implications. If the equilibrium is a saddle point, that means trajectories approach it along one direction and move away along another. So, in the context of predator-prey, does that mean the populations will either spiral towards the equilibrium or away from it? Wait, no, saddle points are unstable, so the equilibrium is unstable.Therefore, the equilibrium point is unstable.Alternatively, maybe I should consider whether the eigenvalues have real parts. Since one eigenvalue is positive and one is negative, the equilibrium is a saddle point, hence unstable.So, in conclusion, the equilibrium point is a saddle point and is unstable.Wait, but let me think again. In the original model, the equilibrium is a center, which is stable in the sense that trajectories are closed orbits around it. But here, with the addition of the constant term, the equilibrium becomes a saddle point, which is unstable. So, the populations won't settle at the equilibrium but will either approach it or move away, depending on the initial conditions.Therefore, the equilibrium is unstable.So, summarizing:1. The modified prey equation is ( frac{dx}{dt} = ax - bxy + e ), leading to an equilibrium at ( x = c/d ) and ( y = (ac + ed)/(bc) ).2. The Jacobian matrix at this equilibrium has eigenvalues with one positive and one negative, indicating the equilibrium is a saddle point and thus unstable.I think that's the conclusion.Final Answer1. The modified system has an equilibrium point at (boxed{left( dfrac{c}{d}, dfrac{ac + ed}{bc} right)}).2. The equilibrium point is unstable, as it is a saddle point.</think>