🧠:<think>Alright, so I have this problem about optimizing a satisfaction score for a software product. The score is given by the function S(x, y) = 3x² + 5xy - y² + 2y + 10. I need to find the values of x and y that maximize this score. Then, in part 2, if x is fixed at 7, I have to find the rate of change of S with respect to y and see if it's increasing or decreasing.Okay, starting with part 1. To maximize a function of two variables, I remember from calculus that I need to find the critical points by taking the partial derivatives with respect to each variable, set them equal to zero, and solve the resulting system of equations. Then, I can check if those points are maxima, minima, or saddle points using the second derivative test.So, first, let me find the partial derivatives of S with respect to x and y.Partial derivative with respect to x:∂S/∂x = d/dx [3x² + 5xy - y² + 2y + 10]= 6x + 5y + 0 + 0 + 0= 6x + 5yPartial derivative with respect to y:∂S/∂y = d/dy [3x² + 5xy - y² + 2y + 10]= 0 + 5x - 2y + 2 + 0= 5x - 2y + 2So, to find critical points, set both partial derivatives equal to zero.Equation 1: 6x + 5y = 0Equation 2: 5x - 2y + 2 = 0Now, I need to solve this system of equations. Let me write them down:1) 6x + 5y = 02) 5x - 2y = -2I can solve this using substitution or elimination. Let's try elimination.First, let's make the coefficients of y the same or opposites. Multiply equation 1 by 2 and equation 2 by 5 to make the coefficients of y 10 and -10.Equation 1 multiplied by 2: 12x + 10y = 0Equation 2 multiplied by 5: 25x - 10y = -10Now, add the two equations together:12x + 10y + 25x - 10y = 0 - 10(12x + 25x) + (10y - 10y) = -1037x = -10So, x = -10 / 37 ≈ -0.2703Hmm, x is negative? But in the problem, x is measured from 0 to 10. So, x can't be negative. That seems odd. Maybe I made a mistake in the calculations.Wait, let me double-check the partial derivatives.∂S/∂x = 6x + 5y. That seems right.∂S/∂y = 5x - 2y + 2. Also correct.So, setting them to zero:6x + 5y = 05x - 2y = -2Wait, maybe I should try substitution instead.From equation 1: 6x + 5y = 0 => 5y = -6x => y = (-6/5)xPlug this into equation 2:5x - 2*(-6/5)x + 2 = 05x + (12/5)x + 2 = 0Convert 5x to 25/5 x to have a common denominator:25/5 x + 12/5 x + 2 = 0(25 + 12)/5 x + 2 = 037/5 x + 2 = 037/5 x = -2x = (-2) * (5/37)x = -10/37 ≈ -0.2703Same result. So, x is negative, which is outside the domain of x (0 to 10). Hmm, that suggests that the critical point is at x ≈ -0.27, which is not within our feasible region. So, does that mean that the maximum occurs on the boundary of the domain?Yes, in optimization problems with constraints, if the critical point is outside the feasible region, the extrema must occur on the boundary. So, since x is between 0 and 10, and y is between 0 and 100, we need to check the function's behavior on the boundaries.But before that, maybe I should check if I did everything correctly. Let me verify the partial derivatives again.Given S(x, y) = 3x² + 5xy - y² + 2y + 10.∂S/∂x: derivative of 3x² is 6x, derivative of 5xy is 5y, rest are constants or y terms, so no x derivative. So, 6x + 5y. Correct.∂S/∂y: derivative of 5xy is 5x, derivative of -y² is -2y, derivative of 2y is 2, rest are constants or x terms. So, 5x - 2y + 2. Correct.So, the critical point is indeed at x ≈ -0.27, y ≈ (from y = (-6/5)x) y ≈ (6/5)(0.27) ≈ 0.324. So, y ≈ 0.324, which is within the y domain (0 to 100). But x is negative, which is outside.Therefore, the maximum must be on the boundary of the domain. So, the boundaries are when x=0, x=10, y=0, y=100, or combinations thereof.But since both x and y are independent variables, we can consider the boundaries as follows:1. Fix x=0, find maximum y in [0,100].2. Fix x=10, find maximum y in [0,100].3. Fix y=0, find maximum x in [0,10].4. Fix y=100, find maximum x in [0,10].5. Also, check the corners: (0,0), (0,100), (10,0), (10,100).But perhaps a better approach is to consider the function S(x, y) and see how it behaves as x and y increase.Looking at S(x, y) = 3x² + 5xy - y² + 2y + 10.Let me analyze the function:- The term 3x² is positive and increases as x increases.- The term 5xy is positive and increases with both x and y.- The term -y² is negative and decreases as y increases.- The term 2y is positive and increases with y.- The constant 10 is just a shift.So, for fixed x, S(x, y) is a quadratic in y: -y² + (5x + 2)y + 3x² + 10.This is a downward opening parabola in y, so it has a maximum at y = -b/(2a) where a = -1, b = 5x + 2.So, y = -(5x + 2)/(2*(-1)) = (5x + 2)/2.So, for each x, the maximum y is (5x + 2)/2. But y is constrained between 0 and 100.So, if (5x + 2)/2 is within [0,100], then that's where the maximum occurs; otherwise, it's at the boundary.Similarly, for fixed y, S(x, y) is a quadratic in x: 3x² + 5xy + (-y² + 2y + 10).This is an upward opening parabola in x, so it has a minimum, not a maximum. Therefore, for fixed y, S(x, y) increases as x increases beyond the vertex. So, to maximize S(x, y), for fixed y, we should set x as large as possible, i.e., x=10.Therefore, the maximum of S(x, y) should occur at x=10, and y as per the maximum for that x.So, let's compute y for x=10.From earlier, y = (5x + 2)/2.Plug x=10: y = (50 + 2)/2 = 52/2 = 26.Since 26 is within [0,100], that's acceptable.Therefore, the maximum occurs at x=10, y=26.Wait, but let me verify this.Alternatively, perhaps I should consider the boundaries.Case 1: x=0.Then S(0, y) = 0 + 0 - y² + 2y + 10 = -y² + 2y + 10.This is a quadratic in y, opening downward. The maximum is at y = -b/(2a) = -2/(2*(-1)) = 1.So, at x=0, maximum at y=1, S= -1 + 2 +10=11.Case 2: x=10.S(10, y) = 3*(100) + 5*10*y - y² + 2y +10 = 300 + 50y - y² + 2y +10 = -y² +52y +310.This is quadratic in y, opening downward. Maximum at y = -b/(2a) = -52/(2*(-1))=26.So, y=26, as before. Then S= -26² +52*26 +310.Calculate that:-26² = -67652*26: 50*26=1300, 2*26=52, total 1352So, -676 +1352 +310 = (1352 -676) +310=676 +310=986.Case 3: y=0.S(x, 0)=3x² +0 -0 +0 +10=3x² +10.This is a quadratic in x, opening upward. So, it has a minimum at x=0, maximum at x=10.So, maximum at x=10, S=3*100 +10=310.Case 4: y=100.S(x,100)=3x² +5x*100 -100² +2*100 +10=3x² +500x -10000 +200 +10=3x² +500x -9790.This is a quadratic in x, opening upward. So, it has a minimum, not a maximum. Therefore, the maximum occurs at x=10.Compute S(10,100)=3*100 +500*10 -9790=300 +5000 -9790=5300 -9790= -4490.That's a very low value.So, comparing the maximums from each boundary:- x=0: S=11- x=10: S=986- y=0: S=310- y=100: S=-4490So, clearly, the maximum is at x=10, y=26, with S=986.Therefore, the answer to part 1 is x=10, y=26.Wait, but let me check if there's a higher value somewhere else.Alternatively, perhaps the maximum is on the edge where y=100, but as we saw, S is negative there, so no.Similarly, on y=0, S is 3x² +10, which is maximized at x=10, giving 310, which is less than 986.So, yes, the maximum is at x=10, y=26.Alternatively, let me think about the function S(x, y). Since for each x, the optimal y is (5x +2)/2, and since this y is within [0,100] for all x in [0,10], because when x=10, y=26, which is within 0-100, and when x=0, y=1, which is also within.Therefore, the maximum occurs at x=10, y=26.So, that's the conclusion.Now, moving on to part 2.The product manager sets x=7. We need to calculate the rate of change of S with respect to y at x=7, which is the partial derivative of S with respect to y at x=7.From earlier, ∂S/∂y =5x -2y +2.So, plug x=7: ∂S/∂y =5*7 -2y +2=35 -2y +2=37 -2y.So, the rate of change is 37 -2y.Now, to determine whether S is increasing or decreasing as y increases, we look at the sign of ∂S/∂y.If ∂S/∂y >0, S is increasing with y.If ∂S/∂y <0, S is decreasing with y.So, 37 -2y >0 when y <18.5.Similarly, 37 -2y <0 when y >18.5.At y=18.5, the rate of change is zero.Therefore, for y <18.5, S is increasing with y.For y >18.5, S is decreasing with y.But in the problem, it just says "calculate the rate of change... and determine whether the satisfaction score is increasing or decreasing as user engagement level increases."But wait, the rate of change is 37 -2y, which is a function of y. So, depending on the value of y, it can be positive or negative.But perhaps the question is asking for the rate of change at a specific y? Or is it asking in general?Wait, the question says: "Calculate the rate of change of the satisfaction score with respect to the user engagement level y at this implementation level, and determine whether the satisfaction score is increasing or decreasing as user engagement level increases."So, it's at x=7, but y is variable. So, the rate of change is 37 -2y, which depends on y. So, the rate of change is a linear function of y, decreasing as y increases.But to determine whether S is increasing or decreasing as y increases, we need to know the sign of ∂S/∂y.But since ∂S/∂y =37 -2y, which can be positive or negative depending on y.But perhaps the question is expecting a general statement, but given that y is a variable, the answer depends on y.Wait, maybe I misinterpreted. Maybe it's asking for the rate of change at a specific point, but the problem doesn't specify a particular y. So, perhaps the answer is that the rate of change is 37 -2y, and S increases when y <18.5 and decreases when y >18.5.Alternatively, maybe the question is expecting the derivative at x=7, regardless of y, so the derivative is 37 -2y, which is a function, and depending on y, it can be positive or negative.But the problem says "determine whether the satisfaction score is increasing or decreasing as user engagement level increases."So, perhaps it's expecting a general statement, but since the derivative is 37 -2y, which can be positive or negative, the answer is that the score increases when y <18.5 and decreases when y >18.5.Alternatively, maybe the question is expecting the derivative at a particular y, but since y isn't specified, perhaps it's just the expression 37 -2y, and the conclusion that it's increasing if y <18.5 and decreasing otherwise.But let me check the problem statement again."Calculate the rate of change of the satisfaction score with respect to the user engagement level y at this implementation level, and determine whether the satisfaction score is increasing or decreasing as user engagement level increases."So, it's at x=7, but y is variable. So, the rate of change is 37 -2y, which is a function of y. So, the rate of change depends on y.Therefore, the answer is that the rate of change is 37 -2y, and the satisfaction score increases when y <18.5 and decreases when y >18.5.Alternatively, if the question is asking for the derivative at a specific y, but since y isn't given, perhaps it's just the expression 37 -2y, and the conclusion that it depends on y.But maybe the question is simpler. Since x=7 is fixed, and we're looking at how S changes with y, the derivative is 37 -2y. So, the rate of change is 37 -2y, and whether it's increasing or decreasing depends on y.But perhaps the question is expecting a numerical value, but without a specific y, we can't compute a numerical rate. So, the answer is that the rate of change is 37 -2y, and the score increases when y <18.5 and decreases when y >18.5.Alternatively, maybe the question is asking for the derivative at x=7, regardless of y, so the derivative is 37 -2y, which is a function, and the conclusion is that the score's behavior depends on y.But perhaps the question is expecting a general answer, like "the rate of change is 37 -2y, and since this can be positive or negative, the score may increase or decrease depending on the value of y."But I think the more precise answer is that the rate of change is 37 -2y, and the score increases when y <18.5 and decreases when y >18.5.So, summarizing:1. The maximum satisfaction score occurs at x=10, y=26.2. When x=7, the rate of change of S with respect to y is 37 -2y, and the score increases when y <18.5 and decreases when y >18.5.But let me double-check part 1 again.We found that the critical point is at x≈-0.27, y≈0.324, which is outside the feasible region. Therefore, the maximum must be on the boundary.We checked the boundaries:- x=0: maximum at y=1, S=11- x=10: maximum at y=26, S=986- y=0: maximum at x=10, S=310- y=100: maximum at x=10, S=-4490So, indeed, the maximum is at x=10, y=26.Therefore, the answers are:1. x=10, y=262. The rate of change is 37 -2y, and the score increases when y <18.5 and decreases when y >18.5.But the question in part 2 says "determine whether the satisfaction score is increasing or decreasing as user engagement level increases." So, perhaps it's expecting a general statement, but since the derivative is 37 -2y, which can be positive or negative, the answer is that it depends on y. If y is less than 18.5, it's increasing; if y is greater than 18.5, it's decreasing.Alternatively, if the question is asking for the derivative at a specific y, but since y isn't given, we can't say definitively whether it's increasing or decreasing without knowing y.But perhaps the question is expecting the derivative expression and the conclusion based on that expression.So, to wrap up:1. The maximum occurs at x=10, y=26.2. The rate of change is 37 -2y, and the score increases when y <18.5 and decreases when y >18.5.</think>

🧠:<think>Okay, so I have this problem about optimizing water distribution in Yemen. There are two parts: first, finding a minimum spanning tree to minimize water loss, and second, developing a linear programming model to maximize water flow from a reservoir. Let me try to break this down step by step.Starting with part 1. The problem is about a graph where cities are nodes and pipelines are edges with weights representing water loss. We need to find an MST. I remember that an MST connects all the nodes with the minimum possible total edge weight without any cycles. So, the goal is to ensure every city is connected with the least water loss.Which algorithm should I use? Krusky's or Prim's? Both are used for MSTs. Krusky's is good for sparse graphs and uses a greedy approach by sorting edges and adding them one by one, avoiding cycles. Prim's starts from a node and grows the MST by adding the smallest edge that connects a new node. Since the problem doesn't specify the graph's density, maybe Krusky's is more general because it can handle any graph.But wait, I should think about the proof. How do I show that Krusky's algorithm gives the optimal solution? I recall that Krusky's works because it uses a greedy approach that always picks the smallest edge that doesn't form a cycle. This is based on the cut property: for any cut of the graph, the MST includes the lightest edge crossing the cut. So, by always choosing the smallest edge that doesn't form a cycle, Krusky's ensures that the total weight is minimized.Alternatively, Prim's algorithm can be proven using the greedy algorithm's correctness, where each step makes a locally optimal choice that leads to a globally optimal solution. But since I'm more comfortable with Krusky's, I'll go with that.So, for part 1, I can say that the problem can be modeled as a weighted graph where each edge's weight is the water loss. The MST will connect all cities with the minimum total water loss. Krusky's algorithm is suitable here because it efficiently finds the MST by sorting edges and using a union-find data structure to avoid cycles. The proof of optimality comes from the cut property, ensuring that each edge added is the minimum possible to connect a new component.Moving on to part 2. Now, each city has a water demand ( d_i ), and each pipeline has a capacity ( c_{ij} ). We need to maximize the total water flow from a main reservoir to all cities, with the total flow not exceeding the reservoir's capacity ( C ). This sounds like a maximum flow problem with demands.Wait, maximum flow usually deals with sending as much flow as possible from source to sink. But here, each city has a demand, so it's more like a flow problem with multiple sinks, each requiring a certain amount of flow. Alternatively, it's a circulation problem where we have to satisfy the demands.But the problem says to maximize the total water flow from the reservoir, given that the total flow cannot exceed ( C ). So, the reservoir is the source, and all cities are nodes that need to receive at least ( d_i ) units of water. But since it's a distribution network, the flow has to go through the pipelines, which have capacities ( c_{ij} ).So, how do I model this as a linear program? Let me think. In linear programming, we can define variables for the flow on each edge. Let ( x_{ij} ) be the flow from city ( i ) to city ( j ). Then, we need to ensure that for each city ( j ), the total inflow minus outflow equals the demand ( d_j ). Except for the reservoir, which is the source.Wait, the reservoir is a single node, let's say node ( s ). So, the flow from the reservoir ( s ) to other cities must satisfy the demands of all cities. Also, the total flow from ( s ) cannot exceed ( C ). So, the total flow out of ( s ) is the sum of ( x_{s j} ) for all ( j ), which should be less than or equal to ( C ).But also, for each city ( i ), the net flow into ( i ) should be equal to ( d_i ). That is, for each city ( i neq s ), the sum of flows into ( i ) minus the sum of flows out of ( i ) equals ( d_i ). For the reservoir ( s ), the net flow is the negative of the total flow, which is ( - sum x_{s j} ), and this should be less than or equal to ( -C ) because the total outflow cannot exceed ( C ).Wait, actually, in flow conservation, for each node ( i ), the inflow minus outflow equals the demand. For the reservoir, the demand is negative because it's supplying water. So, the demand at ( s ) is ( -C ), and the demands at other nodes are ( d_i ).But in the problem statement, it says "maximize the total water flow from the reservoir to all cities, given that the total flow from the reservoir cannot exceed its capacity ( C )." So, maybe the total flow is exactly ( C ), but we have to satisfy all the demands. Hmm, but if the sum of ( d_i ) is greater than ( C ), it's impossible. So, perhaps the objective is to maximize the total flow, which is the sum of ( d_i ) satisfied, but not exceeding ( C ).Wait, the problem says "maximize the total water flow from a main reservoir to all cities, given that the total flow from the reservoir cannot exceed its capacity ( C )." So, the total flow is the sum of flows from the reservoir, which is ( sum x_{s j} ), and we need to maximize this sum, but it cannot exceed ( C ). But also, we have to satisfy the demands of all cities.Wait, maybe it's a standard max flow problem where the source is the reservoir, and each city has a demand, so we need to have a flow that satisfies the demands without exceeding the capacities. But since the reservoir can only send up to ( C ), we need to maximize the total flow sent, which is ( sum x_{s j} ), subject to the constraints that each city ( i ) receives at least ( d_i ), and the pipelines have capacities ( c_{ij} ).But actually, in flow networks, the demands are usually modeled as lower bounds on the flow. So, perhaps we need to set up a flow with lower bounds. Alternatively, we can model it as a standard flow problem by splitting each node into two, one for inflow and one for outflow, connected by an edge with capacity equal to the demand.But maybe a simpler way is to model it as a linear program where we have variables ( x_{ij} ) for each edge, subject to flow conservation, capacity constraints, and the total flow from the reservoir is maximized but not exceeding ( C ).So, let's formalize this.Let me define the nodes as ( V ), with ( s ) being the reservoir. Each city ( i ) has a demand ( d_i ), which must be satisfied. Each pipeline ( (i, j) ) has a capacity ( c_{ij} ). We need to find flows ( x_{ij} ) on each edge such that:1. For each city ( i neq s ), the net flow into ( i ) is at least ( d_i ). That is, ( sum_{j} x_{j i} - sum_{j} x_{i j} geq d_i ).2. For the reservoir ( s ), the net flow out is ( sum_{j} x_{s j} ), which must be less than or equal to ( C ).3. For each edge ( (i, j) ), the flow ( x_{ij} ) must be less than or equal to ( c_{ij} ).4. All flows ( x_{ij} ) must be non-negative.But wait, the objective is to maximize the total water flow from the reservoir, which is ( sum x_{s j} ), subject to the constraints above.But also, we need to ensure that the flows satisfy the demands. So, the constraints are:- For each ( i in V setminus {s} ), ( sum_{j} x_{j i} - sum_{j} x_{i j} geq d_i ).- For ( s ), ( sum_{j} x_{s j} leq C ).- For each edge ( (i, j) ), ( x_{ij} leq c_{ij} ).- ( x_{ij} geq 0 ) for all edges.But is this correct? Because in flow networks, the net flow into a node is equal to the demand. So, for the reservoir ( s ), the net flow is ( - sum x_{s j} ), which should be equal to the negative of the total outflow, which is the total flow we want to maximize. But since the total outflow cannot exceed ( C ), we have ( sum x_{s j} leq C ).But also, for each city ( i ), the net inflow must be at least ( d_i ). So, the constraints are as above.But wait, in linear programming, we can write this as:Maximize ( sum_{j} x_{s j} )Subject to:For each ( i neq s ):( sum_{j} x_{j i} - sum_{j} x_{i j} geq d_i )For ( s ):( sum_{j} x_{s j} leq C )For each edge ( (i, j) ):( x_{ij} leq c_{ij} )And ( x_{ij} geq 0 ) for all edges.But wait, in flow conservation, usually, the net flow into a node is equal to the demand. So, for ( i neq s ), it's ( sum x_{j i} - sum x_{i j} = d_i ). But in our case, since we want to maximize the total flow, perhaps we can relax it to ( geq d_i ), meaning that the cities can receive more than their demand, but we have to ensure they get at least ( d_i ). However, since the reservoir has a limited capacity ( C ), we might not be able to satisfy all demands if the sum of ( d_i ) exceeds ( C ). Therefore, the model should aim to satisfy as much as possible, but the problem says "maximize the total water flow from the reservoir to all cities, given that the total flow from the reservoir cannot exceed its capacity ( C )." So, perhaps the total flow is exactly ( C ), but we have to ensure that all cities receive at least ( d_i ). But if the sum of ( d_i ) is greater than ( C ), it's impossible. Therefore, maybe the problem assumes that the sum of ( d_i ) is less than or equal to ( C ), and we need to route the flow such that each city gets exactly ( d_i ), with the total flow being ( sum d_i ), which is less than or equal to ( C ).Wait, the problem says "maximize the total water flow from the reservoir to all cities, given that the total flow from the reservoir cannot exceed its capacity ( C )." So, perhaps the total flow is the sum of flows from the reservoir, which is ( sum x_{s j} ), and we need to maximize this, but it cannot exceed ( C ). However, we also have to satisfy the demands of each city, meaning that each city ( i ) must receive at least ( d_i ). So, the objective is to maximize ( sum x_{s j} ) subject to ( sum x_{s j} leq C ), and for each city ( i ), the inflow is at least ( d_i ), and the flows on edges do not exceed capacities.But in that case, the problem is to find the maximum possible flow from the reservoir, up to ( C ), that satisfies all the demands. So, the linear program would be:Maximize ( sum_{j} x_{s j} )Subject to:For each ( i neq s ):( sum_{j} x_{j i} - sum_{j} x_{i j} geq d_i )For ( s ):( sum_{j} x_{s j} leq C )For each edge ( (i, j) ):( x_{ij} leq c_{ij} )And ( x_{ij} geq 0 ) for all edges.But wait, this might not capture the flow correctly because the flow conservation for ( s ) is not just an upper bound but also a lower bound. The net flow out of ( s ) is ( sum x_{s j} ), which should be equal to the total flow. But in the constraints, we have ( sum x_{s j} leq C ), but we also need to ensure that the flow satisfies the demands. So, perhaps the model is correct as is.Alternatively, sometimes in flow problems, we set the net flow out of the source to be equal to the total flow, and the net flow into the sink to be equal to that as well. But in this case, each city is a sink with a demand.Wait, perhaps a better way is to model this as a flow network where the reservoir is the source, and each city has a demand, so we can model it by splitting each city into two nodes: one for inflow and one for outflow, connected by an edge with capacity equal to the demand. But that might complicate things.Alternatively, we can use the standard flow conservation constraints with the demands as lower bounds.So, in summary, the linear programming model would have variables ( x_{ij} ) for each edge, representing the flow from ( i ) to ( j ). The objective is to maximize the total flow from the reservoir, which is ( sum x_{s j} ), subject to:1. For each city ( i neq s ), the net inflow is at least ( d_i ): ( sum_{j} x_{j i} - sum_{j} x_{i j} geq d_i )2. For the reservoir ( s ), the total outflow is ( sum x_{s j} leq C )3. For each edge ( (i, j) ), ( x_{ij} leq c_{ij} )4. All ( x_{ij} geq 0 )This should capture the problem correctly. The objective is to maximize the total flow from the reservoir, ensuring that each city gets at least its demand, without exceeding pipeline capacities and the reservoir's capacity.I think that's the correct model. Let me double-check. The total flow from the reservoir is ( sum x_{s j} ), which we want to maximize but cannot exceed ( C ). Each city ( i ) must receive at least ( d_i ), so the net inflow must be at least ( d_i ). The flows on the edges cannot exceed their capacities. And all flows are non-negative. Yes, that seems right.So, to recap:1. Model the problem as a weighted graph, find MST using Krusky's algorithm, which is optimal due to the cut property.2. Formulate the linear program with variables ( x_{ij} ), objective to maximize ( sum x_{s j} ), subject to flow conservation, capacity constraints, and non-negativity.I think that's the solution.</think>

🧠:<think>Okay, so I have this problem where a minister is studying the impact of church attendance and counseling on community well-being. He's modeled the well-being with a function W(t), and there are two sub-problems to solve. Let me tackle them one by one.Starting with Sub-problem 1: The differential equation given is a second-order linear nonhomogeneous differential equation. The equation is:[ frac{d^2W(t)}{dt^2} + 6frac{dW(t)}{dt} + 9W(t) = e^{-3t} ]And the initial conditions are ( W(0) = 2 ) and ( frac{dW(0)}{dt} = -1 ). I need to find the explicit form of W(t).Alright, so for linear differential equations, the general solution is the sum of the homogeneous solution and a particular solution. So, first, I should solve the homogeneous equation:[ frac{d^2W}{dt^2} + 6frac{dW}{dt} + 9W = 0 ]To solve this, I'll find the characteristic equation:[ r^2 + 6r + 9 = 0 ]Let me compute the discriminant: ( D = 36 - 36 = 0 ). So, we have a repeated real root. The root is ( r = frac{-6}{2} = -3 ). Therefore, the homogeneous solution is:[ W_h(t) = (C_1 + C_2 t) e^{-3t} ]Now, I need to find a particular solution ( W_p(t) ) for the nonhomogeneous equation. The right-hand side is ( e^{-3t} ). Hmm, since the homogeneous solution already includes ( e^{-3t} ) and ( t e^{-3t} ), the particular solution can't be just ( A e^{-3t} ) because that's part of the homogeneous solution. So, I need to multiply by t^2 to make it linearly independent. So, let me assume:[ W_p(t) = A t^2 e^{-3t} ]Now, I need to compute the first and second derivatives of W_p(t):First derivative:[ W_p'(t) = A [2t e^{-3t} + t^2 (-3) e^{-3t}] = A e^{-3t} (2t - 3t^2) ]Second derivative:[ W_p''(t) = A [ (2 - 6t) e^{-3t} + (2t - 3t^2)(-3) e^{-3t} ] ]Simplify:First term: ( (2 - 6t) e^{-3t} )Second term: ( -3(2t - 3t^2) e^{-3t} = (-6t + 9t^2) e^{-3t} )Combine them:[ (2 - 6t - 6t + 9t^2) e^{-3t} = (2 - 12t + 9t^2) e^{-3t} ]So, ( W_p''(t) = A (2 - 12t + 9t^2) e^{-3t} )Now, plug ( W_p(t) ), ( W_p'(t) ), and ( W_p''(t) ) into the differential equation:[ W_p'' + 6 W_p' + 9 W_p = e^{-3t} ]Substitute:[ A (2 - 12t + 9t^2) e^{-3t} + 6 A (2t - 3t^2) e^{-3t} + 9 A t^2 e^{-3t} = e^{-3t} ]Factor out ( A e^{-3t} ):[ A e^{-3t} [ (2 - 12t + 9t^2) + 6(2t - 3t^2) + 9t^2 ] = e^{-3t} ]Simplify inside the brackets:First term: 2 - 12t + 9t^2Second term: 12t - 18t^2Third term: 9t^2Combine like terms:Constants: 2t terms: -12t + 12t = 0t^2 terms: 9t^2 - 18t^2 + 9t^2 = 0So, the entire expression simplifies to 2 A e^{-3t} = e^{-3t}Therefore, 2 A = 1 => A = 1/2So, the particular solution is:[ W_p(t) = frac{1}{2} t^2 e^{-3t} ]Therefore, the general solution is:[ W(t) = W_h(t) + W_p(t) = (C_1 + C_2 t) e^{-3t} + frac{1}{2} t^2 e^{-3t} ]Now, apply the initial conditions to find C1 and C2.First, compute W(0) = 2:[ W(0) = (C_1 + C_2 * 0) e^{0} + frac{1}{2} * 0^2 e^{0} = C_1 = 2 ]So, C1 = 2.Next, compute the first derivative of W(t):[ W(t) = (2 + C_2 t) e^{-3t} + frac{1}{2} t^2 e^{-3t} ]Compute W’(t):First, derivative of (2 + C2 t) e^{-3t}:Using product rule: (C2) e^{-3t} + (2 + C2 t)(-3) e^{-3t} = [C2 - 3(2 + C2 t)] e^{-3t}Then, derivative of (1/2 t^2 e^{-3t}):Using product rule: (t) e^{-3t} + (1/2 t^2)(-3) e^{-3t} = [t - (3/2) t^2] e^{-3t}So, overall W’(t):[ [C2 - 6 - 3 C2 t] e^{-3t} + [t - (3/2) t^2] e^{-3t} ]Combine like terms:Constants: C2 - 6t terms: -3 C2 t + tt^2 terms: - (3/2) t^2So, factor out e^{-3t}:[ [ (C2 - 6) + (-3 C2 + 1) t + (-3/2) t^2 ] e^{-3t} ]Now, evaluate W’(0):At t=0, the expression becomes:(C2 - 6) e^{0} = C2 - 6Given that W’(0) = -1, so:C2 - 6 = -1 => C2 = 5Therefore, the explicit form of W(t) is:[ W(t) = (2 + 5t) e^{-3t} + frac{1}{2} t^2 e^{-3t} ]I can factor out e^{-3t}:[ W(t) = e^{-3t} left( 2 + 5t + frac{1}{2} t^2 right) ]Alternatively, I can write it as:[ W(t) = e^{-3t} left( frac{1}{2} t^2 + 5t + 2 right) ]That should be the solution for Sub-problem 1.Moving on to Sub-problem 2: The well-being function is now given by:[ W(t) = W_0(t) + int_0^t C(tau) e^{-k(t - tau)} dtau ]Where ( W_0(t) ) is the solution from Sub-problem 1, ( C(t) = 3 e^{-0.5 t} ), and ( k = 1 ).So, substituting the given values:[ W(t) = W_0(t) + int_0^t 3 e^{-0.5 tau} e^{-1(t - tau)} dtau ]Simplify the exponent in the integral:[ e^{-0.5 tau} e^{-t + tau} = e^{-t} e^{( -0.5 tau + tau )} = e^{-t} e^{0.5 tau} ]So, the integral becomes:[ int_0^t 3 e^{-t} e^{0.5 tau} dtau = 3 e^{-t} int_0^t e^{0.5 tau} dtau ]Compute the integral:[ int e^{0.5 tau} dtau = frac{2}{1} e^{0.5 tau} + C ]So, evaluating from 0 to t:[ 3 e^{-t} [ 2 e^{0.5 t} - 2 e^{0} ] = 3 e^{-t} [ 2 e^{0.5 t} - 2 ] ]Simplify:Factor out 2:[ 3 e^{-t} * 2 [ e^{0.5 t} - 1 ] = 6 e^{-t} (e^{0.5 t} - 1 ) ]Simplify the exponents:[ 6 (e^{-t + 0.5 t} - e^{-t}) = 6 (e^{-0.5 t} - e^{-t}) ]Therefore, the integral part is ( 6 e^{-0.5 t} - 6 e^{-t} ).So, the modified well-being function is:[ W(t) = W_0(t) + 6 e^{-0.5 t} - 6 e^{-t} ]But ( W_0(t) ) is the solution from Sub-problem 1, which is:[ W_0(t) = e^{-3t} left( frac{1}{2} t^2 + 5t + 2 right) ]Therefore, the modified W(t) is:[ W(t) = e^{-3t} left( frac{1}{2} t^2 + 5t + 2 right) + 6 e^{-0.5 t} - 6 e^{-t} ]I think that's the final form. Let me just make sure I didn't make any mistakes in the integral computation.Wait, in the integral, I had:[ int_0^t e^{0.5 tau} dtau = 2 e^{0.5 t} - 2 ]Yes, that's correct because the integral of e^{a x} is (1/a) e^{a x}, so here a = 0.5, so integral is 2 e^{0.5 t}.Then, multiplying by 3 e^{-t}:3 e^{-t} * (2 e^{0.5 t} - 2) = 6 e^{-0.5 t} - 6 e^{-t}Yes, that's correct.So, putting it all together, the modified W(t) is:[ W(t) = e^{-3t} left( frac{1}{2} t^2 + 5t + 2 right) + 6 e^{-0.5 t} - 6 e^{-t} ]I think that's the answer for Sub-problem 2.Final AnswerSub-problem 1: boxed{W(t) = e^{-3t} left( frac{1}{2} t^2 + 5t + 2 right)}Sub-problem 2: boxed{W(t) = e^{-3t} left( frac{1}{2} t^2 + 5t + 2 right) + 6 e^{-0.5 t} - 6 e^{-t}}</think>

🧠:<think>Okay, so I have this problem about a textile worker analyzing VOC emissions from synthetic fabric production. There are two parts: first, calculating the total VOCs emitted in the first 10 hours, and second, figuring out the new total after installing a filtration system. Let me try to work through each step carefully.Starting with the first question. The emission rate is given by E(t) = 10e^{0.3t} kilograms per hour. I need to find the total amount emitted over the first 10 hours. Hmm, I remember that when you have a rate function, the total amount is the integral of that rate over the time period. So, the total VOCs, let's call it V, should be the integral from t=0 to t=10 of E(t) dt.So, V = ∫₀¹⁰ 10e^{0.3t} dt.Alright, integrating an exponential function. The integral of e^{kt} dt is (1/k)e^{kt} + C, right? So, applying that here, the integral of 10e^{0.3t} dt should be 10*(1/0.3)e^{0.3t} evaluated from 0 to 10.Let me write that out:V = 10 / 0.3 [e^{0.3*10} - e^{0.3*0}].Calculating the exponents first. 0.3*10 is 3, so e^3, and e^0 is 1. So,V = (10 / 0.3)(e³ - 1).Let me compute the numerical value. 10 divided by 0.3 is approximately 33.333... So, 33.333*(e³ - 1). I know e³ is roughly 20.0855, so e³ - 1 is about 19.0855.Multiplying that by 33.333: 33.333 * 19.0855. Let me do that step by step.First, 33 * 19 is 627, and 0.333 * 19 is approximately 6.327. So, adding those together, 627 + 6.327 is 633.327. Then, 33.333 * 0.0855 is approximately 2.856. So, adding that to 633.327 gives roughly 636.183.Wait, maybe that's a bit messy. Alternatively, I can compute 33.333 * 19.0855 directly. Let me use a calculator approach.33.333 * 19.0855:First, 33.333 * 19 = 633.327.Then, 33.333 * 0.0855 ≈ 33.333 * 0.08 = 2.66664, and 33.333 * 0.0055 ≈ 0.18333. Adding those gives approximately 2.66664 + 0.18333 ≈ 2.85.So, total is approximately 633.327 + 2.85 ≈ 636.177.So, approximately 636.18 kilograms of VOCs emitted in the first 10 hours.Wait, let me double-check my calculations because I might have made an error in breaking it down. Alternatively, maybe I can compute it more accurately.Alternatively, 33.333 * 19.0855:Compute 33.333 * 19 = 633.327.Compute 33.333 * 0.0855:First, 33.333 * 0.08 = 2.66664.33.333 * 0.0055 = 0.1833315.Adding those gives 2.66664 + 0.1833315 ≈ 2.85.So, total is 633.327 + 2.85 ≈ 636.177, which is approximately 636.18 kg.Alternatively, maybe I can use a calculator for more precision, but since I don't have one, I think 636.18 is a reasonable approximation.So, the total VOCs emitted in the first 10 hours are approximately 636.18 kilograms.Moving on to the second part. The factory installs a filtration system that reduces the emission rate by a factor of e^{-0.1t}. So, the new emission rate is E(t) multiplied by e^{-0.1t}.So, the new emission rate E_new(t) = 10e^{0.3t} * e^{-0.1t} = 10e^{(0.3 - 0.1)t} = 10e^{0.2t}.Wait, that seems too straightforward. Let me confirm: when you multiply exponentials with the same base, you add the exponents. So, e^{0.3t} * e^{-0.1t} = e^{(0.3 - 0.1)t} = e^{0.2t}. So, yes, E_new(t) = 10e^{0.2t}.Therefore, the new total VOCs emitted during the first 10 hours is the integral from 0 to 10 of 10e^{0.2t} dt.So, V_new = ∫₀¹⁰ 10e^{0.2t} dt.Again, integrating e^{kt} is (1/k)e^{kt}, so:V_new = 10 / 0.2 [e^{0.2*10} - e^{0.2*0}].Simplify:0.2*10 is 2, so e², and e^0 is 1.So, V_new = (10 / 0.2)(e² - 1).10 divided by 0.2 is 50, so:V_new = 50*(e² - 1).Compute e²: e² is approximately 7.389056.So, e² - 1 ≈ 6.389056.Multiply by 50: 50 * 6.389056 ≈ 319.4528.So, approximately 319.45 kilograms of VOCs emitted after the filtration system is installed.Wait, let me make sure I didn't make a mistake here. The original emission rate was 10e^{0.3t}, and after the filtration, it's multiplied by e^{-0.1t}, so E_new(t) = 10e^{0.3t} * e^{-0.1t} = 10e^{0.2t}, which is correct.Then, integrating 10e^{0.2t} from 0 to 10:Integral is (10 / 0.2)(e^{0.2*10} - 1) = 50*(e² - 1) ≈ 50*(7.389056 - 1) = 50*6.389056 ≈ 319.4528.Yes, that seems correct.So, the total VOCs after the filtration system are approximately 319.45 kg.Wait, just to make sure, let me check the calculations again.10 divided by 0.2 is indeed 50.e² is approximately 7.389056, so 7.389056 - 1 is 6.389056.50 times 6.389056: 50*6 is 300, 50*0.389056 is approximately 19.4528, so total is 300 + 19.4528 ≈ 319.4528.Yes, that's correct.So, summarizing:1. Without filtration, total VOCs emitted in first 10 hours ≈ 636.18 kg.2. With filtration, total VOCs emitted ≈ 319.45 kg.Wait, just to cross-verify, the filtration system reduces the emission rate by a factor of e^{-0.1t}, which is less than 1, so the new emission rate is lower than the original, which makes sense that the total is lower.Alternatively, maybe I can compute the exact values using more precise exponentials.For the first part, e³ is approximately 20.0855369232.So, e³ - 1 ≈ 19.0855369232.10 / 0.3 is 100/3 ≈ 33.3333333333.So, 33.3333333333 * 19.0855369232 ≈ ?Let me compute 33.3333333333 * 19.0855369232.First, 33 * 19.0855369232 = ?33 * 19 = 627.33 * 0.0855369232 ≈ 33 * 0.0855 ≈ 2.8215.So, total ≈ 627 + 2.8215 ≈ 629.8215.Then, 0.3333333333 * 19.0855369232 ≈ 6.361845641.So, total is approximately 629.8215 + 6.361845641 ≈ 636.183345641.So, approximately 636.1833 kg, which rounds to 636.18 kg as I had before.Similarly, for the second part, e² is approximately 7.38905609893.So, e² - 1 ≈ 6.38905609893.50 * 6.38905609893 ≈ 319.4528049465.So, approximately 319.45 kg.Therefore, my calculations seem consistent.So, to recap:1. Total VOCs emitted in the first 10 hours without filtration: approximately 636.18 kg.2. After installing the filtration system, the total VOCs emitted in the first 10 hours: approximately 319.45 kg.I think that's it. I don't see any mistakes in my reasoning.Final Answer1. The total amount of VOCs emitted during the first 10 hours is boxed{636.18} kilograms.2. The new total amount of VOCs emitted after installing the filtration system is boxed{319.45} kilograms.</think>