🧠:<think>First, I need to calculate the total donations over 10 years. The donations increase by 8% each year, starting with 10,000 in the first year. This forms a geometric series where the first term ( a = 10,000 ) and the common ratio ( r = 1.08 ). The sum of the first 10 terms of this series will give the total donations.Next, I'll determine the future value of the 5,000 investment with continuous compounding. The formula for continuous compound interest is ( A = Pe^{rt} ), where ( P = 5,000 ), ( r = 0.05 ), and ( t = 10 ). Plugging in these values will give the total amount after 10 years.</think>

🧠:<think>First, I'll calculate the cost of upgrading the electrical panel. The base cost is 1,200, and there's an additional 50 per circuit. Since the home requires 15 circuits, the cost for the circuits is 15 multiplied by 50, which equals 750. Adding this to the base cost gives a total of 1,950 for the panel upgrade.Next, I'll determine the rewiring cost. The cost is 7 per linear foot, and the total length needed is 500 feet. Multiplying these gives a rewiring cost of 3,500.Then, I'll calculate the cost for installing the new outlets and fixtures. Each outlet costs 75, and there are 20 outlets, so the total for outlets is 20 times 75, which is 1,500. Each fixture costs 120, and there are 10 fixtures, so the total for fixtures is 10 times 120, equaling 1,200. Adding these together gives a total of 2,700 for outlets and fixtures.Now, I'll sum up all these costs to find the total cost before any discount. Adding the panel upgrade (1,950), rewiring (3,500), and outlets/fixtures (2,700) results in a total of 8,150.Finally, to find the discounted total if the resident pays in full upfront, I'll apply a 10% discount to the total cost. Calculating 10% of 8,150 gives 815. Subtracting this discount from the total cost results in a final amount of 7,335.</think>

🧠:<think>Okay, so I have this problem about a logistics manager trying to optimize delivery schedules for a fleet of trucks. The goal is to minimize total travel time while ensuring continuous communication either with the warehouse or another truck. Let me try to break this down.First, part 1 asks to formulate this as a graph theory problem and express it as an optimization problem. Hmm. So, graph theory often deals with nodes and edges. In this case, the warehouse is at the origin, and the delivery locations are points on a coordinate plane. Each truck starts at the warehouse, visits some unique subset of these locations, and returns. So, each truck's route is a cycle starting and ending at the warehouse.The communication constraint is that each truck must always be within the CB radio range of at least one other truck or the warehouse. The CB radio has a range R, so if two trucks are within R distance of each other, they can communicate. Similarly, a truck is within R of the warehouse if it's within R distance from the origin.So, to model this, I think we can represent each truck's route as a path in a graph. But the key is the communication constraint. This sounds like a connectivity constraint. The entire fleet must form a connected graph where each truck is connected either directly to the warehouse or to another truck within range R.Wait, but each truck is moving along its own route. So, maybe we need to model the communication over time. But that might complicate things. Alternatively, perhaps we can model the routes such that at any point in time, each truck is within R of another truck or the warehouse.But how do we translate this into a graph? Maybe each location (warehouse and delivery points) is a node, and edges exist between nodes if the distance between them is less than or equal to R. Then, the communication constraint requires that the routes form a connected graph where each truck's path is connected to the warehouse or another truck's path.Alternatively, perhaps each truck's route must be connected to the warehouse through a chain of trucks, each within R of the next. So, the entire set of routes must form a connected graph where the warehouse is the root, and each truck is connected through some path of trucks within R.Wait, maybe it's better to think of the problem as a vehicle routing problem with communication constraints. Each truck's route is a tour starting and ending at the warehouse, visiting some subset of delivery points. The communication constraint is that the union of all truck routes must form a connected graph where each node (truck position) is within R of another node.But I'm not sure. Let me think again. The manager needs to ensure that each truck is always in contact with at least one other truck or the warehouse. So, at any point in time, each truck must be within R of another truck or the warehouse.But modeling this over time is complicated because the trucks are moving. So, perhaps we can simplify by considering that the routes must be such that the communication graph is connected at all times. But that might be too strict.Alternatively, maybe the communication graph is built based on the positions of the trucks along their routes. So, for each truck's route, we can model the positions as nodes, and edges exist if two positions are within R. Then, the entire communication graph must be connected.Wait, but each truck's route is a path, so the positions along the route can be considered as a sequence. The communication constraint would require that for each position of a truck, there exists another truck or the warehouse within R.But this seems too vague. Maybe another approach is to model the problem as a graph where nodes are the delivery locations and the warehouse. Edges connect nodes if the distance between them is ≤ R. Then, each truck's route must be a cycle that includes the warehouse and some subset of nodes, and the union of all these cycles must form a connected graph.But that might not capture the continuous communication during travel. Because even if two delivery points are connected by an edge, the truck might be somewhere along the route between them, which might not be within R of another truck.Hmm, this is tricky. Maybe the key is to ensure that the routes are such that the communication graph is connected at all times. So, perhaps the problem can be modeled as a connected vehicle routing problem, where the communication constraints impose that the routes must form a connected network.Alternatively, perhaps the problem can be transformed into a graph where each node represents a truck's position, and edges represent communication links. But since the trucks are moving, this graph is dynamic.Wait, maybe I need to simplify. Let's think of each truck's route as a path in the graph where nodes are delivery locations and the warehouse. The edges have weights equal to the travel time (which is proportional to the Euclidean distance). The communication constraint is that for each node in a truck's route, there must be another node (either the warehouse or another delivery location) within R distance, which is also part of another truck's route.But I'm not sure. Maybe the problem is similar to the connected vehicle routing problem, where the routes must form a connected network. In that case, the problem can be modeled as a connected vehicle routing problem with the communication range constraint.So, in terms of graph theory, the problem is to partition the delivery locations into N subsets, each subset forming a cycle starting and ending at the warehouse, such that the union of these cycles forms a connected graph where each edge is within R distance.But I'm not entirely certain. Maybe I need to look up connected vehicle routing problems, but since I can't do that right now, I'll proceed with my understanding.So, the optimization problem is to find N cycles (one for each truck) starting and ending at the warehouse, covering all delivery locations, with the total travel time minimized. The constraints are:1. Each delivery location is visited by exactly one truck.2. Each truck's route is a cycle starting and ending at the warehouse.3. The communication graph is connected, meaning that for each truck, every point along its route is within R of at least one other truck or the warehouse.Wait, but the third constraint is about the entire route, not just the nodes. So, maybe it's more about the routes being connected in terms of their paths.Alternatively, perhaps the communication graph is built based on the nodes (delivery locations and warehouse), and edges exist if two nodes are within R. Then, the union of all truck routes must form a connected graph in this communication graph.But I think that might not capture the continuous communication during travel. Because even if two delivery points are connected by an edge, the truck might be somewhere along the route between them, which might not be within R of another truck.Hmm, this is getting complicated. Maybe for the purpose of this problem, we can approximate the communication constraint by ensuring that each delivery location is within R of another delivery location or the warehouse, and that the routes are connected in such a way that the communication graph is connected.So, perhaps the problem can be modeled as a graph where nodes are delivery locations and the warehouse, edges connect nodes within R distance, and we need to find N cycles covering all nodes, with the union forming a connected graph, and the total travel time minimized.But I'm not sure if this captures the continuous communication during travel. Maybe it's an acceptable approximation for the problem.So, in terms of optimization, the problem is a vehicle routing problem with the additional constraint that the union of all routes forms a connected graph in the communication range graph. The objective is to minimize the total travel time.Constraints:1. Each delivery location is assigned to exactly one truck.2. Each truck's route starts and ends at the warehouse.3. The union of all truck routes forms a connected graph where each node is connected via edges of distance ≤ R.Wait, but the union of routes forming a connected graph might not necessarily ensure that each truck is within R of another at all times. It might just ensure that the nodes are connected, but not the paths.Alternatively, maybe the communication constraint is that the routes must form a connected network, meaning that there exists a path of trucks where each consecutive truck is within R of each other, connecting back to the warehouse.So, perhaps the problem is similar to the connected vehicle routing problem, where the routes must form a connected network, ensuring that all trucks can communicate through a chain of trucks back to the warehouse.In that case, the problem can be modeled as a connected vehicle routing problem with the communication range R.So, the optimization problem is to find N vehicle routes covering all delivery locations, starting and ending at the warehouse, such that the routes form a connected network where each truck is within R of another truck or the warehouse, and the total travel time is minimized.In terms of graph theory, this would involve finding a connected graph where each node (delivery location) is part of a cycle (truck route), and the communication graph is connected with edges of distance ≤ R.But I'm not entirely sure about the exact formulation. Maybe I should think about it as a combination of vehicle routing and connectivity constraints.So, the formulation would involve variables for each truck's route, ensuring that all delivery locations are covered, the routes are cycles, and the communication graph is connected.In terms of mathematical programming, this would likely be an integer programming problem, where we decide which delivery locations are assigned to which truck, and ensure the connectivity constraint.But I'm not sure how to model the connectivity constraint in integer programming. It might involve ensuring that there's a spanning tree connecting all trucks through the communication graph.Alternatively, perhaps we can model the problem by first constructing the communication graph where nodes are delivery locations and the warehouse, and edges exist if two nodes are within R. Then, the vehicle routing must cover all nodes, and the routes must form a connected subgraph in this communication graph.Wait, that might make sense. So, the communication graph is a graph where nodes are delivery locations and the warehouse, and edges connect nodes within R distance. Then, the vehicle routes must cover all nodes, and the union of the routes must form a connected subgraph in the communication graph.But again, this might not capture the continuous communication during travel, but it's a starting point.So, the optimization problem is to partition the delivery locations into N subsets, each forming a cycle with the warehouse, such that the union of these cycles forms a connected subgraph in the communication graph, and the total travel time is minimized.The constraints are:1. Each delivery location is assigned to exactly one truck.2. Each truck's route is a cycle starting and ending at the warehouse.3. The union of all truck routes forms a connected subgraph in the communication graph (i.e., the communication graph is connected).The objective is to minimize the total travel time, which is the sum of the travel times for each truck's route.Now, for part 2, the problem is similar but with the added detail that travel time is proportional to Euclidean distance. The manager is given a new set of delivery locations, and needs to determine optimal paths for each truck to minimize total travel time, while ensuring that at least one truck is always within R of the warehouse, and all other trucks form a connected subgraph based on their communication capability.So, the constraints are:1. Each truck's route is a cycle starting and ending at the warehouse.2. Each delivery location is visited by exactly one truck.3. At least one truck is always within R of the warehouse (so, its route must stay within R of the warehouse at all times? Or just that it's connected to the warehouse through communication links? Wait, the wording says "at least one truck is always within the CB radio range of the warehouse." So, that truck must be within R of the warehouse at all times. The other trucks must form a connected subgraph based on their communication capability, meaning that each of them is within R of at least one other truck.Wait, so the communication constraints are:- At least one truck is always within R of the warehouse. So, this truck's entire route must be within R of the warehouse. That means its route can't go beyond R distance from the warehouse.- All other trucks must form a connected subgraph based on their communication capability. So, each of these trucks must be within R of at least one other truck (either directly or through a chain), but they don't necessarily have to be within R of the warehouse, as long as they are connected through other trucks.But wait, the problem says "at least one truck is always within the CB radio range of the warehouse, while all other trucks form a connected subgraph based on their communication capability." So, the connected subgraph is among the other trucks, but they don't necessarily have to be connected to the warehouse, except through the first truck.Wait, no, because the first truck is connected to the warehouse, and the other trucks are connected to each other, but to ensure that all trucks are connected to the warehouse, the connected subgraph of the other trucks must be connected to the first truck. Otherwise, they might form a separate connected component.So, perhaps the communication constraint is that the entire set of trucks forms a connected graph, with at least one truck connected directly to the warehouse, and the others connected through the communication links.So, the communication graph must be connected, with the warehouse being part of it through at least one truck.Therefore, the constraints are:1. Each truck's route is a cycle starting and ending at the warehouse.2. Each delivery location is visited by exactly one truck.3. The communication graph is connected, meaning that there's a path of trucks (and the warehouse) where each consecutive pair is within R distance.4. At least one truck is always within R of the warehouse, meaning its entire route is within R of the warehouse.Wait, but if a truck's entire route is within R of the warehouse, then it's always connected to the warehouse. The other trucks just need to be connected through other trucks, but they don't have to be within R of the warehouse directly, as long as they are connected through a chain of trucks back to the warehouse.So, the communication graph must be connected, with the warehouse being connected through at least one truck, and all other trucks connected through their communication links.Therefore, the optimization problem is to find N vehicle routes covering all delivery locations, starting and ending at the warehouse, such that:- At least one truck's entire route is within R of the warehouse.- The communication graph (where nodes are trucks and the warehouse, edges exist if two nodes are within R) is connected.- The total travel time is minimized.But how do we model this? It seems like a combination of vehicle routing and connectivity constraints.In terms of graph theory, perhaps we can model this as a connected vehicle routing problem with a depot (warehouse) and a communication range R. The problem is to find routes for each truck such that:- Each truck starts and ends at the depot.- All delivery locations are visited.- At least one truck's route is entirely within R of the depot.- The communication graph (trucks and depot) is connected, meaning there's a path from the depot to every truck through communication links (distance ≤ R).- The total travel time (sum of Euclidean distances for each truck's route) is minimized.This seems like a variation of the vehicle routing problem with communication constraints.To solve this, advanced optimization techniques like integer programming or combinatorial optimization methods can be used. Specifically, we can model this as an integer linear program where:- Decision variables represent whether a delivery location is assigned to a truck, and the order in which they are visited.- Constraints ensure that each delivery location is assigned to exactly one truck, each truck's route is a cycle, the communication constraints are satisfied, and the total travel time is minimized.However, integer programming can be computationally intensive, especially for large M and N. Therefore, heuristic or metaheuristic methods like genetic algorithms, simulated annealing, or tabu search might be more practical for larger instances.Alternatively, we can decompose the problem into two parts:1. Assign delivery locations to trucks, ensuring that at least one truck is assigned only locations within R of the warehouse.2. For each truck, solve a traveling salesman problem (TSP) to find the optimal route, ensuring that the communication constraints are satisfied.But the communication constraints complicate things because the routes of different trucks must be connected through their communication links.Wait, perhaps we can first construct the communication graph where nodes are delivery locations and the warehouse, and edges exist if two nodes are within R. Then, the problem reduces to finding a connected vehicle routing problem on this graph, where at least one truck's route is entirely within R of the warehouse.But I'm not sure. Maybe another approach is to first ensure that the communication graph is connected, and then solve the vehicle routing problem on it.Alternatively, since the communication constraint requires that the entire set of trucks forms a connected network, perhaps we can model this as a Steiner tree problem, where we need to connect all trucks through the warehouse with minimal additional cost, but I'm not sure.Wait, perhaps the problem can be approached by first ensuring that the communication graph is connected, and then solving the vehicle routing problem on the connected graph.But I'm getting stuck. Maybe I should outline the steps:1. Model the problem as a graph where nodes are delivery locations and the warehouse, edges exist if two nodes are within R.2. Ensure that the graph is connected, with at least one node (truck route) connected directly to the warehouse.3. Partition the nodes into N subsets, each forming a cycle with the warehouse, such that the total travel time is minimized.But I'm not sure how to enforce the communication constraints during the vehicle routing.Alternatively, perhaps we can use a two-step approach:a. First, ensure that the communication graph is connected by selecting a spanning tree that connects all delivery locations through the warehouse, with edges of distance ≤ R.b. Then, partition the spanning tree into N routes, each starting and ending at the warehouse, such that the total travel time is minimized.But this might not capture the optimal routes, as the spanning tree might not correspond to the actual routes taken by the trucks.Alternatively, perhaps we can use a genetic algorithm where each chromosome represents an assignment of delivery locations to trucks, and the fitness function evaluates both the total travel time and the communication constraints.But this is getting too vague. Maybe I should think about the problem in terms of integer programming.Let me define some variables:Let’s denote:- ( M ) delivery locations: ( D = {d_1, d_2, ..., d_M} )- ( N ) trucks: ( T = {t_1, t_2, ..., t_N} )- The warehouse is ( w )- The communication range is ( R )We need to assign each delivery location to a truck, and determine the route for each truck.Let’s define binary variables:- ( x_{ij} ): 1 if delivery location ( i ) is assigned to truck ( j ), 0 otherwise.- ( y_{ijk} ): 1 if truck ( j ) visits location ( i ) immediately before location ( k ), 0 otherwise.But this might get too complex. Alternatively, we can use a flow-based formulation.But perhaps a better approach is to use a set partitioning formulation, where each subset corresponds to a truck's route.However, incorporating the communication constraints into the integer program is challenging.Another idea: To ensure that the communication graph is connected, we can require that there exists a spanning tree connecting all trucks through the warehouse, with edges of distance ≤ R.But how to model this? Maybe we can introduce additional variables to track the communication links.Alternatively, perhaps we can use a constraint that for each truck, there exists a path from the warehouse to that truck through other trucks, each within R.But this is difficult to model in an integer program.Wait, maybe we can model the communication constraints by ensuring that the routes form a connected graph. So, for each pair of trucks, there must be a sequence of trucks where each consecutive pair is within R.But this is similar to ensuring that the communication graph is connected.Alternatively, perhaps we can model the problem as a connected vehicle routing problem, where the communication constraints are part of the problem.In summary, the problem can be formulated as a vehicle routing problem with the additional constraints that:1. At least one truck's route is entirely within R of the warehouse.2. The communication graph (trucks and warehouse) is connected, meaning there's a path of communication links (distance ≤ R) from the warehouse to each truck.The objective is to minimize the total travel time, which is the sum of the Euclidean distances for each truck's route.To solve this, we can use integer programming, where we define variables for the assignment of delivery locations to trucks and the routes, and include constraints for the communication graph connectivity and the R range for at least one truck.However, due to the complexity, especially with large M and N, heuristic methods might be more practical. For example, we can use a genetic algorithm where each chromosome represents a possible assignment of delivery locations to trucks, and the fitness function evaluates both the total travel time and the communication constraints.Alternatively, we can decompose the problem:1. First, assign delivery locations to trucks, ensuring that at least one truck is assigned only locations within R of the warehouse.2. For each truck, solve a TSP to find the optimal route.3. Check if the communication constraints are satisfied (i.e., the communication graph is connected). If not, adjust the assignments and repeat.But this is a heuristic approach and might not find the optimal solution.In conclusion, the problem can be modeled as a vehicle routing problem with communication constraints, formulated as an integer program with specific constraints for connectivity and the R range. However, due to the problem's complexity, heuristic or metaheuristic methods might be more feasible for solving it, especially for larger instances.</think>

🧠:<think>Alright, so I've got this problem about a refugee navigating the asylum process with the help of a counselor. The process has three steps, each with its own probability of success influenced by the counselor's effectiveness, which is denoted by ( g ). The first part asks me to express the overall probability of successfully completing all three steps in terms of ( g ). The second part involves calculating the new overall probability after the counselor improves their effectiveness by 20%, starting from an initial ( g = 0.5 ).Let me start by understanding the first part. The asylum process has three steps: A, B, and C. Each step has a probability of success that depends on ( g ). Specifically, the probabilities are given as:- ( P_A = 0.6 + 0.2g )- ( P_B = 0.5 + 0.3g )- ( P_C = 0.7 + 0.1g )And it's mentioned that the success at each step is independent. So, to find the overall probability of successfully completing the entire process, I need to multiply the probabilities of success at each step because of their independence.So, the overall probability ( P_{text{total}} ) should be:( P_{text{total}} = P_A times P_B times P_C )Substituting the given expressions:( P_{text{total}} = (0.6 + 0.2g) times (0.5 + 0.3g) times (0.7 + 0.1g) )Hmm, that seems straightforward. But maybe I should expand this expression to have it in a more simplified form? Let me see.First, multiply the first two terms:( (0.6 + 0.2g)(0.5 + 0.3g) )Let me compute this multiplication step by step.Multiplying 0.6 by 0.5 gives 0.3.Multiplying 0.6 by 0.3g gives 0.18g.Multiplying 0.2g by 0.5 gives 0.1g.Multiplying 0.2g by 0.3g gives 0.06g².So, adding all these together:0.3 + 0.18g + 0.1g + 0.06g²Combine like terms:0.3 + (0.18g + 0.1g) + 0.06g² = 0.3 + 0.28g + 0.06g²So, the product of the first two steps is ( 0.3 + 0.28g + 0.06g² ).Now, I need to multiply this result by the third term, which is ( 0.7 + 0.1g ).So, let's compute:( (0.3 + 0.28g + 0.06g²) times (0.7 + 0.1g) )Again, I'll break this down term by term.First, multiply 0.3 by 0.7: 0.21Then, 0.3 by 0.1g: 0.03gNext, 0.28g by 0.7: 0.196g0.28g by 0.1g: 0.028g²0.06g² by 0.7: 0.042g²0.06g² by 0.1g: 0.006g³Now, adding all these together:0.21 + 0.03g + 0.196g + 0.028g² + 0.042g² + 0.006g³Combine like terms:Constant term: 0.21g terms: 0.03g + 0.196g = 0.226gg² terms: 0.028g² + 0.042g² = 0.07g²g³ term: 0.006g³So, putting it all together:( 0.21 + 0.226g + 0.07g² + 0.006g³ )Therefore, the overall probability ( P_{text{total}} ) is:( P_{text{total}} = 0.006g³ + 0.07g² + 0.226g + 0.21 )Wait, let me double-check my calculations to make sure I didn't make any errors.Starting with the first multiplication:( (0.6 + 0.2g)(0.5 + 0.3g) )0.6*0.5 = 0.30.6*0.3g = 0.18g0.2g*0.5 = 0.1g0.2g*0.3g = 0.06g²Adding: 0.3 + 0.18g + 0.1g + 0.06g² = 0.3 + 0.28g + 0.06g². That seems correct.Then, multiplying by ( 0.7 + 0.1g ):First term: 0.3*0.7 = 0.210.3*0.1g = 0.03g0.28g*0.7 = 0.196g0.28g*0.1g = 0.028g²0.06g²*0.7 = 0.042g²0.06g²*0.1g = 0.006g³Adding all:0.21 + 0.03g + 0.196g + 0.028g² + 0.042g² + 0.006g³Combine:0.21 + (0.03 + 0.196)g + (0.028 + 0.042)g² + 0.006g³Which is 0.21 + 0.226g + 0.07g² + 0.006g³. That looks correct.So, the overall probability is a cubic polynomial in terms of ( g ):( P_{text{total}} = 0.006g³ + 0.07g² + 0.226g + 0.21 )Alternatively, I can write it as:( P_{text{total}} = 0.006g^3 + 0.07g^2 + 0.226g + 0.21 )I think that's the expression for the overall probability in terms of ( g ).Moving on to the second part. The counselor can improve their effectiveness ( g ) by attending a special training program, which increases ( g ) by 20%. Initially, ( g = 0.5 ). So, I need to compute the new ( g ) after the 20% increase and then plug that into the overall probability expression.First, let's find the new ( g ). A 20% increase on ( g = 0.5 ) is calculated as:( Delta g = 0.20 times 0.5 = 0.10 )So, the new ( g ) is:( g_{text{new}} = 0.5 + 0.10 = 0.6 )Alternatively, since it's a 20% increase, we can compute it as:( g_{text{new}} = 0.5 times (1 + 0.20) = 0.5 times 1.20 = 0.6 )Either way, ( g_{text{new}} = 0.6 ).Now, I need to compute ( P_{text{total}} ) with ( g = 0.6 ). I can use the expanded polynomial expression I found earlier or compute each probability separately and then multiply them. Let me try both ways to verify.First, using the expanded polynomial:( P_{text{total}} = 0.006g³ + 0.07g² + 0.226g + 0.21 )Substituting ( g = 0.6 ):Compute each term:1. ( 0.006 times (0.6)^3 )2. ( 0.07 times (0.6)^2 )3. ( 0.226 times 0.6 )4. ( 0.21 )Calculating each:1. ( (0.6)^3 = 0.216 ) So, ( 0.006 times 0.216 = 0.001296 )2. ( (0.6)^2 = 0.36 ) So, ( 0.07 times 0.36 = 0.0252 )3. ( 0.226 times 0.6 = 0.1356 )4. ( 0.21 ) remains as is.Adding all these together:0.001296 + 0.0252 + 0.1356 + 0.21Let's add step by step:First, 0.001296 + 0.0252 = 0.026496Then, 0.026496 + 0.1356 = 0.162096Next, 0.162096 + 0.21 = 0.372096So, approximately 0.3721, or 37.21%.Alternatively, let me compute each probability separately and then multiply them.Given ( g = 0.6 ):Compute ( P_A = 0.6 + 0.2g = 0.6 + 0.2*0.6 = 0.6 + 0.12 = 0.72 )Compute ( P_B = 0.5 + 0.3g = 0.5 + 0.3*0.6 = 0.5 + 0.18 = 0.68 )Compute ( P_C = 0.7 + 0.1g = 0.7 + 0.1*0.6 = 0.7 + 0.06 = 0.76 )Now, multiply these together:( P_{text{total}} = 0.72 times 0.68 times 0.76 )Let me compute this step by step.First, multiply 0.72 and 0.68:0.72 * 0.68Let me compute 72 * 68 first:72 * 60 = 432072 * 8 = 576So, 4320 + 576 = 4896Since both numbers have two decimal places, the result is 0.4896.So, 0.72 * 0.68 = 0.4896Now, multiply this by 0.76:0.4896 * 0.76Again, compute 4896 * 76:First, 4896 * 70 = 342,720Then, 4896 * 6 = 29,376Adding together: 342,720 + 29,376 = 372,096Since the original numbers have four decimal places (0.4896 has four, 0.76 has two; total six decimal places), but wait, actually:Wait, 0.4896 is four decimal places, 0.76 is two decimal places, so total six decimal places. However, 0.4896 * 0.76 = 0.372096Wait, let me verify:0.4896* 0.76--------Multiply 0.4896 by 0.7: 0.34272Multiply 0.4896 by 0.06: 0.029376Add them together: 0.34272 + 0.029376 = 0.372096Yes, that's correct. So, 0.4896 * 0.76 = 0.372096So, ( P_{text{total}} = 0.372096 ), which is approximately 0.3721 or 37.21%.Comparing this with the result from the expanded polynomial, which was approximately 0.3721, they match. So, that's a good consistency check.Therefore, the new overall probability after the counselor's effectiveness increases to 0.6 is approximately 37.21%.Wait, but let me think again. The initial ( g ) was 0.5, and after a 20% increase, it becomes 0.6. So, plugging into the original expression, we get approximately 37.21% success probability.Just to make sure, let me compute the original expression with ( g = 0.5 ) as well, just to see the difference.Compute ( P_{text{total}} ) when ( g = 0.5 ):Using the expanded polynomial:( P_{text{total}} = 0.006*(0.5)^3 + 0.07*(0.5)^2 + 0.226*(0.5) + 0.21 )Compute each term:1. ( 0.006*(0.125) = 0.00075 )2. ( 0.07*(0.25) = 0.0175 )3. ( 0.226*0.5 = 0.113 )4. ( 0.21 )Adding them up:0.00075 + 0.0175 = 0.018250.01825 + 0.113 = 0.131250.13125 + 0.21 = 0.34125So, approximately 0.34125 or 34.125%.Alternatively, computing each probability separately:( P_A = 0.6 + 0.2*0.5 = 0.6 + 0.1 = 0.7 )( P_B = 0.5 + 0.3*0.5 = 0.5 + 0.15 = 0.65 )( P_C = 0.7 + 0.1*0.5 = 0.7 + 0.05 = 0.75 )Multiply them together:0.7 * 0.65 = 0.4550.455 * 0.75 = 0.34125Which is 34.125%, same as before.So, with ( g = 0.5 ), the overall probability is 34.125%, and after increasing ( g ) to 0.6, it becomes approximately 37.21%. So, the improvement is about 3.085% increase in the overall success probability.Just to ensure that my polynomial expansion was correct, let me compute ( P_{text{total}} ) with ( g = 0.6 ) using the original expression without expanding:( P_{text{total}} = (0.6 + 0.2*0.6) * (0.5 + 0.3*0.6) * (0.7 + 0.1*0.6) )Compute each term:1. ( 0.6 + 0.2*0.6 = 0.6 + 0.12 = 0.72 )2. ( 0.5 + 0.3*0.6 = 0.5 + 0.18 = 0.68 )3. ( 0.7 + 0.1*0.6 = 0.7 + 0.06 = 0.76 )Multiply them:0.72 * 0.68 = 0.48960.4896 * 0.76 = 0.372096Which is the same as before. So, the polynomial expansion was correct.Therefore, the overall probability after the counselor's effectiveness increases to 0.6 is approximately 0.3721, or 37.21%.I think I've covered all the steps and double-checked my calculations. It seems consistent.Final Answer1. The overall probability is boxed{0.006g^3 + 0.07g^2 + 0.226g + 0.21}.2. The new overall probability is boxed{0.3721}.</think>