🧠:<think>Alright, so I have this problem about optimizing a software project by refactoring its codebase. It's split into two parts. Let me try to tackle them one by one.Starting with the first part: Given a directed acyclic graph (DAG) representing the dependency structure of modules, each with a certain number of lines of code, I need to find the critical path. The critical path is the longest path in the DAG, right? Because that would determine the minimum time required to complete the project since all dependencies must be completed before moving on to the next module.So, the problem is essentially to find the longest path in a DAG. I remember that in graph theory, the longest path problem is NP-hard for general graphs, but since we have a DAG, it can be solved efficiently using topological sorting.Let me think about how to model this. Each node has a weight, which is the number of lines of code. The edges represent dependencies, meaning that a module can't be started until all its dependencies are completed. So, the critical path is the sequence of modules where each module depends on the previous one, and the sum of their weights is the maximum possible.To find this, I can perform a topological sort on the DAG. Once the nodes are ordered topologically, I can then compute the longest path from the start node to each node by considering each node in the topological order and updating the maximum path length for its neighbors.Mathematically, for each node ( v_i ), the longest path ending at ( v_i ) is the maximum of the longest paths ending at all its predecessors plus the weight of ( v_i ). So, if I denote ( L(v_i) ) as the length of the longest path ending at ( v_i ), then:[L(v_i) = w(v_i) + max_{v_j in text{predecessors}(v_i)} L(v_j)]If a node has no predecessors, its longest path is just its own weight. After computing ( L(v_i) ) for all nodes, the maximum value among all ( L(v_i) ) will be the length of the critical path.So, the algorithm steps would be:1. Perform a topological sort on the DAG.2. Initialize the longest path lengths for all nodes to their own weights.3. For each node in topological order, iterate through its neighbors and update their longest path lengths if a longer path is found through the current node.4. The maximum value in the longest path array is the critical path length.That makes sense. Now, moving on to the second part. The maintainer wants to assign contributors to refactor different modules. Each contributor has a skill level ( s_k ), and the time taken by contributor ( C_k ) to refactor module ( v_i ) is proportional to ( frac{w(v_i)}{s_k} ). The goal is to assign contributors to modules to minimize the overall refactoring time.This sounds like an assignment problem where we have to assign tasks (refactoring modules) to workers (contributors) such that the total time is minimized. Since each contributor can only be assigned to one module, and each module needs to be refactored by exactly one contributor, this is a classic bipartite matching problem.To model this, I can represent it as a bipartite graph where one set is the modules and the other set is the contributors. The edges between them have weights equal to ( frac{w(v_i)}{s_k} ), representing the time it would take contributor ( C_k ) to refactor module ( v_i ). The problem then reduces to finding a minimum weight matching in this bipartite graph.This is known as the Assignment Problem, and it can be solved using the Hungarian Algorithm. The Hungarian Algorithm efficiently finds the minimum cost matching in a bipartite graph, which in this case would correspond to the minimum total refactoring time.Let me outline the steps:1. Create a cost matrix where rows represent modules and columns represent contributors. The entry at (i, k) is ( frac{w(v_i)}{s_k} ).2. Apply the Hungarian Algorithm to this matrix to find the assignment that minimizes the total cost (total refactoring time).3. The result will be an assignment of each module to a contributor such that the sum of the times is minimized.Wait, but what if the number of modules and contributors isn't the same? The problem statement says "a set of potential contributors who can each be assigned to refactor different modules." So, I assume the number of contributors is equal to the number of modules, otherwise, it's not a one-to-one assignment. If there are more contributors than modules, we might have some contributors not assigned, but the problem says "each can be assigned to refactor different modules," implying a bijection.So, assuming the number of modules equals the number of contributors, the cost matrix is square, and the Hungarian Algorithm applies directly.Alternatively, if the numbers are different, we might need to adjust the problem, but since it's not specified, I think we can proceed under the assumption that it's a square matrix.Therefore, the mathematical model is a bipartite graph with weighted edges, and the solution is the minimum weight matching, found via the Hungarian Algorithm.Let me recap:1. For the critical path, model it as the longest path in a DAG, solved via topological sorting and dynamic programming.2. For the assignment problem, model it as a bipartite graph with costs ( frac{w(v_i)}{s_k} ), solved via the Hungarian Algorithm.I think that covers both parts. I should make sure I'm not missing any constraints or details. For the first part, the DAG is given, so topological sort is straightforward. For the second part, the assignment is one-to-one, so the Hungarian Algorithm is appropriate.Final Answer1. The critical path problem is solved by finding the longest path in the DAG using topological sorting and dynamic programming. The algorithm involves topologically sorting the graph and then computing the longest path for each node based on its predecessors. The critical path is the longest such path.2. The optimal assignment problem is modeled as a minimum weight bipartite matching problem, where the cost of assigning contributor ( C_k ) to module ( v_i ) is ( frac{w(v_i)}{s_k} ). This is solved using the Hungarian Algorithm.The final answers are:1. The critical path is found using topological sorting and dynamic programming, resulting in the longest path in the DAG. The answer is boxed{text{Longest Path in DAG}}.2. The optimal assignment is determined by the Hungarian Algorithm applied to the cost matrix, resulting in the minimum total refactoring time. The answer is boxed{text{Hungarian Algorithm}}.</think>

🧠:<think>Alright, so I'm trying to help this screenwriter with their math problem. Let me read through the problem again to make sure I understand it.First, there are three characters, each with their own quadratic time functions: ( T_1(t) ), ( T_2(t) ), and ( T_3(t) ). Each of these is a quadratic equation, so they have the form ( a_i t^2 + b_i t + c_i ) where ( i = 1, 2, 3 ). The screenwriter wants the total screen time ( T(t) = T_1(t) + T_2(t) + T_3(t) ) to be a linear function of ( t ). So, part 1 is about figuring out the conditions on the coefficients ( a_i ), ( b_i ), and ( c_i ) for this to happen.Okay, so if each ( T_i(t) ) is quadratic, then adding them together would normally result in another quadratic function. But the screenwriter wants the total to be linear. That means the quadratic terms must cancel out. So, the coefficients of ( t^2 ) in the total function must sum to zero.Let me write that out:( T(t) = T_1(t) + T_2(t) + T_3(t) = (a_1 + a_2 + a_3) t^2 + (b_1 + b_2 + b_3) t + (c_1 + c_2 + c_3) )For this to be a linear function, the coefficient of ( t^2 ) must be zero. So,( a_1 + a_2 + a_3 = 0 )That's the first condition. The other coefficients, ( b_i ) and ( c_i ), can be anything because they contribute to the linear and constant terms, respectively, which are allowed in a linear function.So, for part 1, the condition is that the sum of the quadratic coefficients ( a_1 + a_2 + a_3 = 0 ).Now, moving on to part 2. The screenwriter wants that at the end of 10 years, the total screen time for each character is equal. Also, the total screen time for each character over 10 years must sum to 300 minutes. Wait, that might need some clarification.Wait, does it mean that each character's total screen time over 10 years is equal, and each of those totals is 300 minutes? Or does it mean that the total screen time across all three characters is 300 minutes, with each character having equal time?Hmm, let's read it again: "the total screen time for each character over the 10 years must sum to 300 minutes." Hmm, that wording is a bit confusing. Maybe it means that each character's total screen time over 10 years is 300 minutes? Or that the sum of all three characters' screen times over 10 years is 300 minutes?Wait, the original problem says: "the total screen time for each character should be equal. Given that the total screen time for each character over the 10 years must sum to 300 minutes..."Wait, maybe it's saying that each character's total screen time over 10 years is equal, and the sum of all three is 300 minutes. So, each character would have 100 minutes total? Because 300 divided by 3 is 100.Alternatively, maybe it's saying that each character's total is 300 minutes, but that would make the total 900 minutes, which seems high. But the wording is a bit unclear.Wait, let's parse it: "the total screen time for each character should be equal. Given that the total screen time for each character over the 10 years must sum to 300 minutes."Hmm, maybe it's saying that each character's total is 300 minutes. So, each ( T_i(10) = 300 ). But that would mean each character has 300 minutes over 10 years, so the total would be 900 minutes. But the problem says "the total screen time for each character over the 10 years must sum to 300 minutes." Hmm, maybe it's that the sum of all three characters' screen times is 300 minutes, and each character's total is equal. So each would be 100 minutes.I think that makes more sense because 300 divided by 3 is 100. So, each character's total screen time over 10 years is 100 minutes.So, for each character, ( T_i(10) = 100 ) minutes.Additionally, from part 1, we have the condition that ( a_1 + a_2 + a_3 = 0 ).So, we need to find quadratic functions ( T_1(t) ), ( T_2(t) ), and ( T_3(t) ) such that:1. ( T_1(t) + T_2(t) + T_3(t) ) is linear, so ( a_1 + a_2 + a_3 = 0 ).2. For each ( i ), ( T_i(10) = 100 ).Additionally, since each ( T_i(t) ) is a quadratic function, we might need more conditions to determine them uniquely. But the problem doesn't specify anything else, so perhaps we can assume some symmetry or choose coefficients that satisfy the conditions.Wait, but each ( T_i(t) ) is a quadratic function, so each has three coefficients ( a_i ), ( b_i ), ( c_i ). We have three functions, so 9 variables in total. From part 1, we have one condition: ( a_1 + a_2 + a_3 = 0 ). From part 2, we have three conditions: ( T_1(10) = 100 ), ( T_2(10) = 100 ), ( T_3(10) = 100 ). So, that's four conditions. We need more conditions to solve for the 9 variables.Hmm, maybe the screenwriter wants the functions to be symmetric in some way? Or perhaps each character's screen time starts at the same point? Or maybe they have the same initial screen time?Wait, the problem doesn't specify anything else, so maybe we can make some assumptions. For simplicity, perhaps each ( T_i(t) ) has the same quadratic coefficient but opposite signs? Wait, but we already have ( a_1 + a_2 + a_3 = 0 ). So, maybe two of them have the same ( a ) and the third has ( -2a ) to balance it out.Alternatively, maybe each ( T_i(t) ) is a quadratic function that starts at the same point, say ( c_1 = c_2 = c_3 = c ), and their linear coefficients are arranged in a way that the total is linear.But without more information, it's hard to determine the exact functions. Maybe the problem expects a general form rather than specific functions. Let me think.Wait, the problem says "find the specific quadratic functions," so maybe we need to assign specific values. Let's see.We have:1. ( a_1 + a_2 + a_3 = 0 )2. For each ( i ), ( T_i(10) = a_i (10)^2 + b_i (10) + c_i = 100 )So, for each ( i ):( 100a_i + 10b_i + c_i = 100 )That's three equations.We also need to ensure that ( T(t) ) is linear, so ( a_1 + a_2 + a_3 = 0 ).But we have 9 variables and only 4 equations. So, we need to make some assumptions or set some variables to specific values.Perhaps, for simplicity, we can assume that each ( T_i(t) ) has the same linear coefficient ( b_i = b ) and the same constant term ( c_i = c ). Then, we can solve for ( a_i ), ( b ), and ( c ).Let me try that.Assume ( b_1 = b_2 = b_3 = b ) and ( c_1 = c_2 = c_3 = c ).Then, from the total being linear, ( a_1 + a_2 + a_3 = 0 ).From each ( T_i(10) = 100 ):( 100a_i + 10b + c = 100 ) for each ( i ).So, for each ( i ), ( 100a_i = 100 - 10b - c ).But since ( a_1 + a_2 + a_3 = 0 ), we can write:( a_1 + a_2 + a_3 = 0 )But each ( a_i = (100 - 10b - c)/100 ). Wait, no, because ( 100a_i = 100 - 10b - c ), so ( a_i = (100 - 10b - c)/100 ).But if all ( a_i ) are equal, then ( 3a = 0 ), so ( a = 0 ). But then ( 100a_i = 0 = 100 - 10b - c ), so ( 10b + c = 100 ).But if ( a_i = 0 ), then each ( T_i(t) ) is linear, which contradicts the requirement that each is quadratic. So, this approach doesn't work.Alternatively, maybe not all ( a_i ) are equal, but they sum to zero. Let's suppose that two of them are equal and the third is the negative of their sum.Let me set ( a_1 = a ), ( a_2 = a ), then ( a_3 = -2a ).Then, for each ( i ), ( 100a_i + 10b_i + c_i = 100 ).So, for ( i = 1 ): ( 100a + 10b_1 + c_1 = 100 )For ( i = 2 ): ( 100a + 10b_2 + c_2 = 100 )For ( i = 3 ): ( 100(-2a) + 10b_3 + c_3 = 100 )So, we have:1. ( 100a + 10b_1 + c_1 = 100 )2. ( 100a + 10b_2 + c_2 = 100 )3. ( -200a + 10b_3 + c_3 = 100 )Now, we have three equations, but we also have the freedom to choose ( b_1, b_2, b_3, c_1, c_2, c_3 ). To simplify, maybe set ( b_1 = b_2 = b_3 = b ) and ( c_1 = c_2 = c_3 = c ). Let's see if that works.Then, equations become:1. ( 100a + 10b + c = 100 )2. ( 100a + 10b + c = 100 )3. ( -200a + 10b + c = 100 )So, equations 1 and 2 are the same, and equation 3 is different.Subtract equation 1 from equation 3:( (-200a + 10b + c) - (100a + 10b + c) = 100 - 100 )Simplify:( -300a = 0 )So, ( a = 0 ). But again, this leads to ( a = 0 ), which makes each ( T_i(t) ) linear, which contradicts the quadratic requirement.Hmm, so assuming all ( b_i ) and ( c_i ) are equal doesn't work. Maybe we need to let them vary.Alternatively, perhaps each ( T_i(t) ) has the same linear coefficient ( b ), but different constants ( c_i ). Let's try that.Let ( b_1 = b_2 = b_3 = b ).Then, equations:1. ( 100a_1 + 10b + c_1 = 100 )2. ( 100a_2 + 10b + c_2 = 100 )3. ( 100a_3 + 10b + c_3 = 100 )And ( a_1 + a_2 + a_3 = 0 ).Let me denote ( S = a_1 + a_2 + a_3 = 0 ).From each equation, we can express ( c_i = 100 - 100a_i - 10b ).So, ( c_1 = 100 - 100a_1 - 10b )( c_2 = 100 - 100a_2 - 10b )( c_3 = 100 - 100a_3 - 10b )Now, since ( S = 0 ), we can write ( a_3 = -a_1 - a_2 ).So, ( c_3 = 100 - 100(-a_1 - a_2) - 10b = 100 + 100a_1 + 100a_2 - 10b )But we still have three variables ( a_1, a_2, b ) and three equations (from the ( c_i ) expressions), but it's getting complicated.Alternatively, maybe we can set ( a_1 = a ), ( a_2 = a ), ( a_3 = -2a ), and then solve for ( b ) and ( c_i ).So, let's try that.Set ( a_1 = a ), ( a_2 = a ), ( a_3 = -2a ).Then, for each ( i ):1. ( 100a + 10b_1 + c_1 = 100 )2. ( 100a + 10b_2 + c_2 = 100 )3. ( -200a + 10b_3 + c_3 = 100 )Now, let's assume that ( b_1 = b_2 = b_3 = b ). Then:1. ( 100a + 10b + c_1 = 100 )2. ( 100a + 10b + c_2 = 100 )3. ( -200a + 10b + c_3 = 100 )From equations 1 and 2, ( c_1 = c_2 ). Let's denote ( c_1 = c_2 = c ).Then, equation 3 becomes ( -200a + 10b + c_3 = 100 ).But from equation 1: ( c = 100 - 100a - 10b ).So, ( c_3 = 100 + 200a - 10b ).But we also have ( c_3 = 100 - 100a_3 - 10b = 100 - 100(-2a) - 10b = 100 + 200a - 10b ), which matches.So, we have:( c_1 = c_2 = 100 - 100a - 10b )( c_3 = 100 + 200a - 10b )Now, we need to choose ( a ) and ( b ) such that the functions are quadratic, so ( a neq 0 ).But we have some freedom here. Let's choose ( a = 1 ) for simplicity.Then, ( a_1 = 1 ), ( a_2 = 1 ), ( a_3 = -2 ).From equation 1:( 100(1) + 10b + c = 100 )So, ( 100 + 10b + c = 100 ) => ( 10b + c = 0 ) => ( c = -10b )Similarly, ( c_3 = 100 + 200(1) - 10b = 300 - 10b )But we need to ensure that the screen time is non-negative for all ( t ) in [0,10]. So, ( T_i(t) geq 0 ) for all ( t ).Let's check for ( T_3(t) = -2t^2 + b t + c_3 )With ( c_3 = 300 - 10b ), so ( T_3(t) = -2t^2 + b t + 300 - 10b )We need this to be non-negative for ( t ) from 0 to 10.Similarly, ( T_1(t) = t^2 + b t + c = t^2 + b t -10b )And ( T_2(t) = t^2 + b t -10b )So, let's analyze ( T_1(t) ):( T_1(t) = t^2 + b t -10b )We need this to be non-negative for ( t in [0,10] ).Similarly, ( T_3(t) = -2t^2 + b t + 300 -10b )We need this to be non-negative for ( t in [0,10] ).Let's find the minimum of ( T_1(t) ). Since it's a quadratic opening upwards, the minimum is at ( t = -b/(2*1) = -b/2 ). But since ( t geq 0 ), if the vertex is at negative ( t ), the minimum on [0,10] is at ( t=0 ).At ( t=0 ), ( T_1(0) = 0 + 0 -10b = -10b ). For this to be non-negative, ( -10b geq 0 ) => ( b leq 0 ).Similarly, for ( T_3(t) ), which is a downward opening quadratic, the maximum is at ( t = -b/(2*(-2)) = b/4 ). So, the minimum occurs at the endpoints ( t=0 ) or ( t=10 ).At ( t=0 ), ( T_3(0) = 0 + 0 + 300 -10b = 300 -10b ). Since ( b leq 0 ), ( 300 -10b geq 300 ), which is positive.At ( t=10 ), ( T_3(10) = -200 + 10b + 300 -10b = 100 ), which is positive.So, as long as ( b leq 0 ), ( T_1(t) ) and ( T_3(t) ) are non-negative.Now, let's choose ( b = 0 ) for simplicity.Then, ( c = -10b = 0 ), and ( c_3 = 300 -10b = 300 ).So, the functions become:( T_1(t) = t^2 + 0*t + 0 = t^2 )( T_2(t) = t^2 + 0*t + 0 = t^2 )( T_3(t) = -2t^2 + 0*t + 300 = -2t^2 + 300 )Now, let's check if these satisfy the conditions.First, total screen time ( T(t) = t^2 + t^2 + (-2t^2 + 300) = 300 ), which is a constant function, hence linear (with slope 0). So, that's good.Each ( T_i(10) ):( T_1(10) = 100 )( T_2(10) = 100 )( T_3(10) = -200 + 300 = 100 )Perfect, each is 100 minutes.But wait, ( T_1(t) = t^2 ) starts at 0 and goes up to 100. ( T_3(t) = -2t^2 + 300 ) starts at 300 and goes down to 100. ( T_2(t) = t^2 ) is the same as ( T_1(t) ).But is this a good representation? It might be a bit simplistic, but mathematically, it satisfies all the conditions.Alternatively, maybe we can choose ( b ) to be negative to have some linear growth or decay.Let me try ( b = -10 ).Then, ( c = -10*(-10) = 100 )( c_3 = 300 -10*(-10) = 300 + 100 = 400 )So, functions:( T_1(t) = t^2 -10t + 100 )( T_2(t) = t^2 -10t + 100 )( T_3(t) = -2t^2 -10t + 400 )Check ( T(t) = T_1 + T_2 + T_3 = (t^2 -10t + 100) + (t^2 -10t + 100) + (-2t^2 -10t + 400) = (1+1-2)t^2 + (-10-10-10)t + (100+100+400) = 0t^2 -30t + 600 ). So, ( T(t) = -30t + 600 ), which is linear. Good.Each ( T_i(10) ):( T_1(10) = 100 -100 + 100 = 100 )( T_2(10) = 100 -100 + 100 = 100 )( T_3(10) = -200 -100 + 400 = 100 )Good.Now, check if ( T_i(t) geq 0 ) for all ( t in [0,10] ).For ( T_1(t) = t^2 -10t + 100 ). The discriminant is ( 100 - 400 = -300 ), so it doesn't cross zero, and since the coefficient of ( t^2 ) is positive, it's always positive.For ( T_3(t) = -2t^2 -10t + 400 ). Let's find its roots:( -2t^2 -10t + 400 = 0 )Multiply by -1: ( 2t^2 +10t -400 = 0 )Divide by 2: ( t^2 +5t -200 = 0 )Solutions: ( t = [-5 ± sqrt(25 + 800)]/2 = [-5 ± sqrt(825)]/2 ≈ [-5 ± 28.72]/2 )Positive root: (23.72)/2 ≈ 11.86, which is beyond our interval [0,10]. So, at t=10, it's 100, which is positive, and since it's a downward opening parabola, it's positive throughout [0,10].So, this works.But the problem says "find the specific quadratic functions," so maybe this is acceptable. However, the problem might expect a different approach.Alternatively, perhaps each character's screen time starts at the same point and ends at the same point, with different growth rates.Wait, but in the case where ( a_1 = a_2 = 1 ) and ( a_3 = -2 ), we have two characters whose screen time increases quadratically and one whose decreases quadratically, balancing out to a linear total.But maybe the screenwriter wants each character's screen time to be symmetric in some way.Alternatively, perhaps each ( T_i(t) ) is symmetric around t=5, but that might complicate things.Alternatively, maybe each character's screen time is a quadratic that starts and ends at the same value, but that would require ( T_i(0) = T_i(10) ), which would mean ( c_i = 100a_i + 10b_i + c_i ), leading to ( 100a_i + 10b_i = 0 ), so ( 10a_i + b_i = 0 ). But that's an additional condition.But the problem doesn't specify that, so perhaps it's not necessary.Given that, I think the simplest solution is to set ( a_1 = a_2 = 1 ), ( a_3 = -2 ), ( b_1 = b_2 = b_3 = 0 ), and ( c_1 = c_2 = 0 ), ( c_3 = 300 ). This gives:( T_1(t) = t^2 )( T_2(t) = t^2 )( T_3(t) = -2t^2 + 300 )Which satisfies all conditions: total is linear (constant 300), each ( T_i(10) = 100 ), and each is quadratic.Alternatively, with ( b = -10 ), we have more varied functions but still satisfying the conditions.But since the problem asks for specific functions, perhaps the simplest is the one with ( b = 0 ).So, final answer for part 2:( T_1(t) = t^2 )( T_2(t) = t^2 )( T_3(t) = -2t^2 + 300 )But let me double-check:Total ( T(t) = t^2 + t^2 + (-2t^2 + 300) = 300 ), which is linear.Each ( T_i(10) = 100 ), as required.Yes, that works.Alternatively, if we want the total to be a non-constant linear function, we can choose ( b neq 0 ). For example, with ( b = -10 ), the total is ( -30t + 600 ), which is linear and non-constant.But unless specified, either could be acceptable. Since the problem doesn't specify the form of the linear function, just that it's linear, both are valid.But perhaps the simplest is the constant total, so I'll go with that.So, summarizing:1. The condition is ( a_1 + a_2 + a_3 = 0 ).2. The specific functions are ( T_1(t) = t^2 ), ( T_2(t) = t^2 ), ( T_3(t) = -2t^2 + 300 ).</think>

🧠:<think>Alright, so I have this problem where a parent volunteer is organizing a fundraising event for a STEM club's field trip. The event involves selling handmade crafts and baked goods. The crafts are priced at 15 each, and the baked goods are 10 each. The goal is to raise at least 1,500. Let me break this down into two sub-problems as given.Sub-problem 1: Formulating an Inequality and Finding Minimum SalesFirst, I need to represent the number of crafts sold as ( x ) and the number of baked goods sold as ( y ). Since each craft brings in 15 and each baked good brings in 10, the total money raised from crafts would be ( 15x ) and from baked goods would be ( 10y ). The fundraising goal is to raise at least 1,500. So, the total money from both crafts and baked goods should be greater than or equal to 1,500. That translates to the inequality:[ 15x + 10y geq 1500 ]Okay, that makes sense. Now, I need to graph this inequality. To graph it, I can treat it as a linear equation first:[ 15x + 10y = 1500 ]To graph this, I can find the intercepts. - When ( x = 0 ), ( 10y = 1500 ) which means ( y = 150 ).- When ( y = 0 ), ( 15x = 1500 ) which means ( x = 100 ).So, the line passes through the points (0, 150) and (100, 0). Since the inequality is ( geq ), the feasible region is above this line. I should shade the area above the line to represent all combinations of ( x ) and ( y ) that meet or exceed the fundraising goal.Now, to determine the minimum number of crafts and baked goods needed, I think we're looking for the smallest integer values of ( x ) and ( y ) that satisfy the inequality. Since both ( x ) and ( y ) can't be negative, we're dealing with non-negative integers.One approach is to find the intercepts as I did before. If we sell only crafts, we need at least 100 crafts. If we sell only baked goods, we need at least 150 baked goods. But maybe a combination of both would require fewer total items sold.Wait, actually, the question is asking for the minimum number of crafts and baked goods. It might be interpreted as the minimum total number of items, but the wording says "minimum number of crafts and baked goods that need to be sold." Hmm, maybe it's asking for the minimum number in each category? Or perhaps the minimum total?Wait, let me read it again: "Determine the minimum number of crafts and baked goods that need to be sold to meet or exceed the fundraising goal."Hmm, it's a bit ambiguous. It could mean the minimum total number of items, or it could mean the minimum number for each. But since it's two variables, it's more likely that they want the minimum number for each, but given that it's a linear inequality, there isn't a single minimum unless we fix one variable.Wait, perhaps they want the minimum number of crafts and baked goods, meaning the smallest ( x ) and ( y ) such that ( 15x + 10y geq 1500 ). But without additional constraints, this could be any combination where ( x ) and ( y ) are as small as possible.But actually, to minimize the total number of items sold, we should maximize the contribution per item, which is higher for crafts (15) than baked goods (10). So, to minimize the total number, we should sell as many crafts as possible.Wait, but the question isn't explicitly asking for the minimum total number, just the minimum number of crafts and baked goods. Maybe it's asking for the minimum number for each, but that doesn't make much sense because if you sell more crafts, you can sell fewer baked goods, and vice versa.Alternatively, perhaps it's asking for the minimum number of each that must be sold regardless of the other. But that would require more information, like if there's a limit on how many of each can be sold.Wait, maybe I'm overcomplicating. Since the problem is about formulating the inequality and shading the feasible region, and then determining the minimum number. So, perhaps the minimum number is the intercepts, meaning if you only sell crafts, you need 100, if you only sell baked goods, you need 150. So, the minimum number of crafts is 100, and the minimum number of baked goods is 150. But that seems a bit off because if you sell a combination, you might need fewer of each.Wait, let me think again. If I want to minimize the number of crafts, I need to maximize the number of baked goods, but since baked goods bring in less money, you'd need more of them. Conversely, to minimize the number of baked goods, you need to maximize crafts.But the question is asking for the minimum number of crafts and baked goods. Maybe it's asking for the smallest number of each that, when combined, meet the goal. But without knowing the exact number of each, it's hard to say.Wait, perhaps it's asking for the minimum total number of items. So, to minimize ( x + y ) subject to ( 15x + 10y geq 1500 ). That would be an optimization problem.Let me set that up. Let me denote ( T = x + y ). We want to minimize ( T ) subject to ( 15x + 10y geq 1500 ) and ( x, y geq 0 ).To minimize ( T ), we can use the method of linear programming. The feasible region is the area above the line ( 15x + 10y = 1500 ). The minimum ( T ) will occur at a vertex of the feasible region.The vertices are at (0, 150) and (100, 0). So, calculating ( T ) at these points:- At (0, 150): ( T = 0 + 150 = 150 )- At (100, 0): ( T = 100 + 0 = 100 )So, the minimum total number of items is 100, which is achieved by selling 100 crafts and 0 baked goods.But wait, the question is asking for the minimum number of crafts and baked goods, not the total. So, perhaps it's asking for the minimum number of each, but that doesn't make much sense because you can have different combinations.Alternatively, maybe it's asking for the minimum number of each such that the total is at least 1500. But without more constraints, you can have any combination where ( x ) and ( y ) satisfy the inequality.Wait, maybe the question is just asking for the intercepts, meaning the minimum number of crafts is 100 and the minimum number of baked goods is 150. So, if you only sell crafts, you need at least 100, and if you only sell baked goods, you need at least 150.But perhaps the answer expects the minimum number of each, meaning the smallest ( x ) and ( y ) such that ( 15x + 10y geq 1500 ). But without knowing the exact values, it's hard to say. Maybe the question is just asking for the intercepts as the minimum for each.Alternatively, maybe it's asking for the minimum number of each when combined. For example, the smallest ( x ) and ( y ) such that their combination meets the goal. But without knowing the exact values, it's tricky.Wait, perhaps the answer is simply that the minimum number of crafts is 100 and the minimum number of baked goods is 150. So, if you don't sell any baked goods, you need 100 crafts, and if you don't sell any crafts, you need 150 baked goods. That seems to make sense.So, for Sub-problem 1, the inequality is ( 15x + 10y geq 1500 ), the graph would have the line connecting (0,150) and (100,0), shaded above. The minimum number of crafts is 100 and the minimum number of baked goods is 150.Sub-problem 2: Calculating the Participation FeeNow, moving on to Sub-problem 2. The bus rental costs 350, and the club is taking 40 students. They charge each student a participation fee to cover the bus rental. Additionally, 10% of the funds raised from crafts and baked goods are allocated to cover the bus rental.So, the total funds allocated to the bus rental come from two sources: the participation fees and 10% of the fundraising.Let me denote the participation fee per student as ( p ). Then, the total participation fees collected would be ( 40p ).Additionally, 10% of the total funds raised from crafts and baked goods is allocated to the bus. The total funds raised from crafts and baked goods is ( 15x + 10y ). So, 10% of that is ( 0.10(15x + 10y) ).The total amount allocated to the bus rental is the sum of the participation fees and the 10% from fundraising:[ 40p + 0.10(15x + 10y) = 350 ]But wait, we also know from Sub-problem 1 that ( 15x + 10y geq 1500 ). So, the minimum total funds raised is 1500. Therefore, the 10% of that is 150.So, substituting that into the equation:[ 40p + 150 = 350 ]Solving for ( p ):[ 40p = 350 - 150 ][ 40p = 200 ][ p = 200 / 40 ][ p = 5 ]So, the minimum participation fee per student needed is 5.Wait, but let me double-check. The 10% is from the funds raised, which is at least 1500, so 10% is at least 150. Therefore, the participation fees need to cover the remaining 200, which with 40 students is 5 each. That seems correct.Alternatively, if the funds raised are more than 1500, the participation fee could be less, but since we're asked for the minimum participation fee needed to cover the entire cost, we have to assume the minimum funds raised, which is 1500. Therefore, the calculation is correct.So, the participation fee per student is 5.Final AnswerSub-problem 1: The inequality is ( 15x + 10y geq 1500 ). The minimum number of crafts is boxed{100} and the minimum number of baked goods is boxed{150}.Sub-problem 2: The minimum participation fee per student is boxed{5} dollars.</think>

🧠:<think>Alright, so I have this problem about a river valley shaped like a parabola, and I need to figure out the new equation after some erosion and then calculate the change in width after 5 years. Let me try to break this down step by step.First, problem 1: The original parabola is given by ( y = ax^2 + bx + c ) with ( a = 0.5 ), ( b = 0 ), and ( c = 0 ). So plugging those in, the original equation is ( y = 0.5x^2 ). That makes sense; it's a simple upward-opening parabola with vertex at the origin.Now, due to erosion, the parabola has shifted. The new equation is given as ( y = a(x - d)^2 + e ). They mentioned that the vertex has shifted 3 units to the right and 2 units up. Hmm, okay, so in the vertex form of a parabola, ( y = a(x - h)^2 + k ), the vertex is at (h, k). So if it's shifted 3 units to the right, that means h is 3, and 2 units up, so k is 2. Therefore, the new equation should be ( y = 0.5(x - 3)^2 + 2 ). Let me double-check that. If I expand this, it should give me the standard form. Let's see:( y = 0.5(x^2 - 6x + 9) + 2 )( y = 0.5x^2 - 3x + 4.5 + 2 )( y = 0.5x^2 - 3x + 6.5 )Wait, but the original equation was ( y = 0.5x^2 ). So the new equation is indeed different. So I think that's correct. The vertex is now at (3, 2), so the equation is ( y = 0.5(x - 3)^2 + 2 ).Okay, moving on to problem 2. The river is flowing at a rate of 4 meters per year, and we need to calculate the change in the width of the valley after 5 years. The river is modeled as the line ( y = 0 ), so we need to find where the parabola intersects the river, which is where ( y = 0 ). The width of the valley at that point is the distance between the two intersection points.First, let's find the current width. The current equation is ( y = 0.5(x - 3)^2 + 2 ). To find the intersection with ( y = 0 ), set ( 0 = 0.5(x - 3)^2 + 2 ).So, ( 0.5(x - 3)^2 + 2 = 0 )Subtract 2: ( 0.5(x - 3)^2 = -2 )Multiply both sides by 2: ( (x - 3)^2 = -4 )Wait, hold on. That can't be right because a square can't be negative. So does that mean the parabola doesn't intersect the river? That doesn't make sense because the river is flowing through the valley. Maybe I made a mistake.Wait, let me check the equation again. The original parabola was ( y = 0.5x^2 ), which intersects the x-axis (river) at x = 0, but since it's a parabola opening upwards, it only touches the origin. But after shifting, the vertex is at (3, 2), so the parabola is above the river. So, does that mean the river is at a lower elevation, and the valley is above it? Hmm, maybe I misunderstood.Wait, maybe the river is at the bottom of the valley, so the parabola represents the valley's cross-section, with the river at the lowest point. If the vertex is shifted up, does that mean the river is still at y=0, but the valley is now higher? Or maybe the river is eroding the sides, making the valley wider.Wait, perhaps I need to think differently. Maybe the parabola represents the river's course, and the width is the distance between the two points where the river meets the valley walls. So if the river is eroding, the parabola is getting wider, meaning the roots are moving further apart.But in the original equation, ( y = 0.5x^2 ), the river is at y=0, which intersects only at x=0. That's just a single point, which doesn't make sense for a valley. Maybe the original equation is meant to be a downward-opening parabola? Because if it's upward-opening, the river would only intersect at one point.Wait, let me go back to the problem statement. It says the river flows through the valley that is shaped like a parabola. So maybe the parabola is actually a cross-section of the valley, with the river at the bottom. So if it's a downward-opening parabola, the river would be at the vertex, and the valley would widen as you go away from the river.But in the original equation, ( y = 0.5x^2 ), which is upward-opening. So perhaps I need to consider that the parabola is actually the cross-section of the valley, with the river at the bottom, so maybe it's a downward-opening parabola. But the original equation given is upward-opening. Hmm, this is confusing.Wait, maybe I need to flip the parabola. If the original equation is ( y = 0.5x^2 ), and the river is at y=0, then the valley is above the river, but that doesn't make much sense. Maybe the equation is actually ( y = -0.5x^2 ), which would open downward, making the river at the bottom. But the problem says ( a = 0.5 ), so it's positive. Hmm.Alternatively, maybe the parabola is oriented such that the river is at the vertex, and the sides slope upward. So the original equation is ( y = 0.5x^2 ), which is a U-shaped valley with the river at the bottom (vertex at (0,0)). Then, due to erosion, the parabola shifts, changing the width.Wait, but in the first problem, the vertex shifted 3 units to the right and 2 units up, so the new vertex is at (3, 2). So the river is still at y=0, but the parabola is now shifted so that the vertex is above the river. So the parabola doesn't intersect the river anymore? That can't be right because the river is flowing through the valley.Wait, maybe I need to adjust my understanding. Perhaps the parabola is the cross-section of the valley, with the river at the bottom, so the parabola opens downward. But in the original equation, it's opening upward. Maybe the problem is using a different orientation.Alternatively, perhaps the parabola is the river's path, and the valley is the area around it. So the river is at y=0, and the valley is the area above it, shaped like a parabola. So the original equation is ( y = 0.5x^2 ), which is a U-shape above the river. Then, due to erosion, the parabola shifts, making the valley wider.Wait, but if the parabola is shifting, the intersection points with the river (y=0) would change. So initially, the original parabola ( y = 0.5x^2 ) intersects the river at x=0, but that's just a single point. That doesn't make sense for a valley. Maybe the original equation is supposed to be a downward-opening parabola, so ( y = -0.5x^2 ), which would intersect the river at two points.But the problem states ( a = 0.5 ), so it's positive. Hmm, this is confusing. Maybe I need to proceed with the given information.So, original equation: ( y = 0.5x^2 ). After shifting, it's ( y = 0.5(x - 3)^2 + 2 ). Now, the river is at y=0, so to find the current width, we need to find where ( y = 0 ) intersects the new parabola.So set ( 0 = 0.5(x - 3)^2 + 2 ). As I did before, this leads to ( (x - 3)^2 = -4 ), which has no real solutions. That means the parabola doesn't intersect the river. That can't be right because the river is flowing through the valley. So perhaps the parabola is actually opening downward, so the equation should be ( y = -0.5x^2 ), but the problem says ( a = 0.5 ). Hmm.Wait, maybe the parabola is shifted such that it now intersects the river. Let me think. If the vertex is at (3, 2), and the parabola opens upward, then the river is below the vertex, so the parabola doesn't intersect the river. That doesn't make sense. So maybe the parabola should open downward. Let me check the problem statement again.Problem 1 says the original equation is ( y = ax^2 + bx + c ) with ( a = 0.5 ), so it's upward-opening. After shifting, it's ( y = a(x - d)^2 + e ). So if the vertex is shifted 3 units right and 2 units up, the new equation is ( y = 0.5(x - 3)^2 + 2 ), which is still upward-opening. So the parabola doesn't intersect the river at y=0. That seems contradictory because the river is supposed to be flowing through the valley.Wait, maybe the river is not at y=0 in the original equation. Maybe the river is at the vertex, so in the original equation, the river is at (0,0), and the valley is above it. Then, after shifting, the river is still at y=0, but the valley's vertex is now at (3,2). So the parabola is above the river, meaning the river is at the bottom, but the valley is now shifted. So the river is still flowing through the valley, but the valley's shape has changed.Wait, but if the parabola is above the river, then the river is at y=0, and the valley is the area above it. So the width of the valley at the river level would be the distance between the two points where the parabola intersects the river. But in the original equation, ( y = 0.5x^2 ), the river is at y=0, so the intersection is only at x=0, which is a single point. That doesn't make sense for a valley.I think I'm getting confused here. Maybe I need to consider that the parabola is the cross-section of the valley, with the river at the bottom. So the original parabola is ( y = -0.5x^2 ), opening downward, with the river at y=0, which intersects at x=0. But the problem says ( a = 0.5 ), so it's positive. Hmm.Alternatively, maybe the parabola is the river's path, and the valley is the area around it. So the river is at y=0, and the valley is the area above it, shaped like a parabola. So the original equation is ( y = 0.5x^2 ), which is above the river. Then, due to erosion, the parabola shifts, making the valley wider.Wait, but if the parabola is shifting, the intersection points with the river (y=0) would change. So initially, the original parabola ( y = 0.5x^2 ) intersects the river only at x=0, which is a single point. That doesn't make sense for a valley. So maybe the original equation is actually a downward-opening parabola, so ( y = -0.5x^2 ), which would intersect the river at two points, x=0. But that's still just one point.Wait, no. If it's a downward-opening parabola, the vertex is at the top, so the river would be at the bottom, which is y approaching negative infinity. That doesn't make sense either.I think I need to clarify this. Maybe the parabola is the cross-section of the valley, with the river at the bottom, so the parabola opens upward, with the river at y=0. So the original equation is ( y = 0.5x^2 ), which intersects the river at x=0, but that's just a single point. That doesn't make sense for a valley. So perhaps the original equation is actually ( y = -0.5x^2 ), opening downward, with the river at y=0, which would intersect at x=0, but again, just one point.Wait, maybe the original equation is ( y = 0.5x^2 ), and the river is not at y=0, but at some other level. But the problem says the river is at y=0. Hmm.Alternatively, maybe the parabola is the river's course, and the valley is the area around it. So the river is at y=0, and the valley is the area above it, shaped like a parabola. So the original equation is ( y = 0.5x^2 ), which is above the river. Then, due to erosion, the parabola shifts, making the valley wider.But if the parabola is shifting, the intersection points with the river (y=0) would change. So initially, the original parabola ( y = 0.5x^2 ) intersects the river only at x=0, which is a single point. That doesn't make sense for a valley. So maybe the original equation is actually a downward-opening parabola, so ( y = -0.5x^2 ), which would intersect the river at two points, x=0, but that's still just one point.Wait, no. If it's a downward-opening parabola, the vertex is at the top, so the river would be at the bottom, which is y approaching negative infinity. That doesn't make sense either.I think I'm stuck here. Maybe I need to proceed with the given information, even if it seems contradictory.So, original equation: ( y = 0.5x^2 ). After shifting, it's ( y = 0.5(x - 3)^2 + 2 ). Now, the river is at y=0, so to find the current width, we need to find where ( y = 0 ) intersects the new parabola.Set ( 0 = 0.5(x - 3)^2 + 2 ). As before, this leads to ( (x - 3)^2 = -4 ), which has no real solutions. So the parabola doesn't intersect the river. That can't be right because the river is flowing through the valley. So maybe the parabola is actually opening downward, so ( y = -0.5(x - 3)^2 + 2 ). Let me try that.Set ( 0 = -0.5(x - 3)^2 + 2 )Multiply both sides by -1: ( 0 = 0.5(x - 3)^2 - 2 )Add 2: ( 2 = 0.5(x - 3)^2 )Multiply by 2: ( 4 = (x - 3)^2 )Take square roots: ( x - 3 = pm 2 )So x = 3 + 2 = 5 or x = 3 - 2 = 1So the intersection points are at x=1 and x=5. Therefore, the width of the valley is the distance between these two points, which is 5 - 1 = 4 units. So the current width is 4 units.Wait, but the original equation was ( y = 0.5x^2 ), which if it's opening downward, would be ( y = -0.5x^2 ). But the problem says ( a = 0.5 ), so it's positive. Hmm. Maybe the problem intended the parabola to open downward, but stated a positive a. That might be a mistake in the problem.Alternatively, maybe the river is eroding the sides, making the parabola wider, so the width increases over time. But in the current equation, the parabola doesn't intersect the river, so the width is zero. That doesn't make sense.Wait, maybe I need to consider that the parabola is shifted such that it now intersects the river. So if the original parabola was ( y = 0.5x^2 ), and it's shifted 3 units right and 2 units up, making it ( y = 0.5(x - 3)^2 + 2 ), which doesn't intersect y=0. So perhaps the river is eroding the sides, making the parabola wider, which would mean the parabola is shifting downward, not upward.Wait, that might make sense. If the river is eroding the sides, the valley would widen, which could mean the parabola is shifting downward, making the vertex lower. But the problem says the vertex shifted 3 units to the right and 2 units up. Hmm.Alternatively, maybe the parabola is shifting in such a way that it intersects the river at two points, even though the vertex is above the river. But as we saw, with the vertex at (3,2), the parabola doesn't intersect y=0.Wait, maybe the parabola is not just shifted, but also stretched or compressed. But the problem only mentions shifting, not scaling. So the a value remains 0.5.I think I need to proceed with the assumption that the parabola is opening downward, even though the problem says a=0.5. So the new equation is ( y = -0.5(x - 3)^2 + 2 ). Then, the intersection points are at x=1 and x=5, giving a width of 4 units.Now, for the second part, the river is eroding at a rate of 4 meters per year, widening the valley symmetrically. So after 5 years, the width will have increased by 4 m/year * 5 years = 20 meters. Wait, but the width is currently 4 units, so does that mean each year the width increases by 4 meters? Or is the rate 4 meters per year in total?Wait, the problem says the river is flowing at a rate of 4 meters per year. I think that refers to the rate of erosion, which is causing the valley to widen. So if the valley is widening symmetrically, the rate at which each side is moving away from the river is 4 meters per year. So each year, the width increases by 8 meters (4 meters on each side). Therefore, after 5 years, the width would increase by 8 * 5 = 40 meters.Wait, but the original width was 4 units. Are these units meters? The problem doesn't specify, but since the rate is given in meters per year, I think the units are consistent. So the original width is 4 meters, and after 5 years, it's 4 + 40 = 44 meters.But wait, let me think again. If the river is eroding at 4 meters per year, does that mean the width increases by 4 meters per year, or 8 meters per year (4 on each side)? The problem says the erosion widens the valley symmetrically, so it's 4 meters per year on each side, making the total increase 8 meters per year.Therefore, after 5 years, the change in width is 8 * 5 = 40 meters.But wait, the original width was 4 units. If each unit is a meter, then the original width is 4 meters. After 5 years, it's 4 + 40 = 44 meters. So the change is 40 meters.Alternatively, if the rate is 4 meters per year total, then after 5 years, the width increases by 4 * 5 = 20 meters. So the change is 20 meters.I think the problem says the river is flowing at a rate of 4 meters per year, which is the rate of erosion. So if it's eroding 4 meters per year on each side, the total increase is 8 meters per year. But if it's 4 meters per year total, then it's 4 meters per year.The problem says "the river is currently flowing through the valley at a rate of 4 meters per year." Hmm, that might refer to the flow rate, not the erosion rate. Wait, no, the problem says "the river is flowing through the valley at a rate of 4 meters per year. Assuming the erosion process widens the valley symmetrically..."Wait, maybe the rate of 4 meters per year is the rate at which the valley is widening. So each year, the width increases by 4 meters. So after 5 years, the width increases by 20 meters.But earlier, I calculated the current width as 4 meters. So after 5 years, it's 4 + 20 = 24 meters. So the change is 20 meters.But I'm not sure. The problem says "the river is flowing through the valley at a rate of 4 meters per year." That might refer to the flow velocity, but then it says "the erosion process widens the valley symmetrically." So maybe the rate of widening is 4 meters per year.Alternatively, perhaps the rate of erosion is 4 meters per year on each side, so total widening is 8 meters per year.I think the problem is a bit ambiguous, but given that it says "the river is flowing through the valley at a rate of 4 meters per year," and then mentions the erosion process widens the valley, I think it's safer to assume that the rate of widening is 4 meters per year total, not per side. So after 5 years, the width increases by 4 * 5 = 20 meters.Therefore, the change in width is 20 meters.Wait, but earlier, I assumed the parabola was opening downward, which might not be correct. Let me try to clarify.If the original parabola is ( y = 0.5x^2 ), which is upward-opening, and the river is at y=0, then the valley is above the river, and the river is just a line at the bottom. But the parabola doesn't intersect the river except at the vertex, which is a single point. So the width is zero, which doesn't make sense.Alternatively, if the parabola is opening downward, ( y = -0.5x^2 ), then it intersects the river at two points, x=0, but that's still just one point. Wait, no, if it's opening downward, the vertex is at the top, so the river is at the bottom, which is y approaching negative infinity. That doesn't make sense either.I think the problem might have a mistake in the orientation of the parabola. Maybe the original equation should be ( y = -0.5x^2 ), opening downward, with the river at y=0, intersecting at two points. Then, after shifting, the new equation is ( y = -0.5(x - 3)^2 + 2 ), which would intersect the river at two points.Let me try that. So original equation: ( y = -0.5x^2 ). After shifting 3 units right and 2 units up, it's ( y = -0.5(x - 3)^2 + 2 ).Set ( y = 0 ):( 0 = -0.5(x - 3)^2 + 2 )Multiply by -1: ( 0 = 0.5(x - 3)^2 - 2 )Add 2: ( 2 = 0.5(x - 3)^2 )Multiply by 2: ( 4 = (x - 3)^2 )Take square roots: ( x - 3 = pm 2 )So x = 5 and x = 1.Therefore, the current width is 5 - 1 = 4 meters.Now, the river is eroding at a rate of 4 meters per year, widening the valley symmetrically. So each year, the width increases by 4 meters. After 5 years, the width increases by 4 * 5 = 20 meters. So the new width is 4 + 20 = 24 meters. Therefore, the change in width is 20 meters.But wait, if the parabola is opening downward, the vertex is at (3, 2), and the river is at y=0, which is below the vertex. So the parabola intersects the river at x=1 and x=5, giving a width of 4 meters. If the river is eroding, the parabola would shift further away from the river, making the width larger.But how does the erosion affect the equation? If the parabola is shifting, the vertex would move further away, but the problem says the river is eroding at a rate of 4 meters per year, which is causing the valley to widen. So perhaps the parabola is shifting downward, making the vertex lower, which would increase the width.Wait, but the problem says the vertex shifted 3 units to the right and 2 units up in problem 1. So in problem 2, the river is eroding, causing the valley to widen, which would mean the parabola is shifting further away from the river, perhaps shifting downward.But the problem doesn't specify any further shifts; it just asks for the change in width after 5 years due to the erosion rate. So maybe the rate of 4 meters per year is the rate at which the width is increasing.Therefore, after 5 years, the width increases by 4 * 5 = 20 meters.So the change in width is 20 meters.But wait, in problem 1, the width was 4 meters. After 5 years, it's 24 meters. So the change is 20 meters.Alternatively, if the rate is 4 meters per year on each side, making the total increase 8 meters per year, then after 5 years, the width increases by 40 meters, making the total width 44 meters, and the change is 40 meters.But the problem says "the river is flowing through the valley at a rate of 4 meters per year. Assuming the erosion process widens the valley symmetrically..."I think the rate of 4 meters per year is the total increase in width per year, not per side. So each year, the width increases by 4 meters. Therefore, after 5 years, the width increases by 20 meters.So the change in width is 20 meters.But I'm still not entirely sure. Maybe I should calculate it based on the rate of erosion affecting the parabola's equation.If the river is eroding at a rate of 4 meters per year, and the valley is widening symmetrically, that means each side is moving away from the river at 2 meters per year. So the width increases by 4 meters per year (2 meters on each side).Therefore, after 5 years, the width increases by 4 * 5 = 20 meters.So the change in width is 20 meters.Yes, that makes sense. So the answer is 20 meters.</think>