answer：First, I need to determine the total yards gained by Tom's passes. Tom completed 18 passes, and each pass averaged 7 yards. To find the total yards, I'll multiply the number of passes by the average yards per pass.Next, I'll calculate the points from Tom's touchdowns. Each touchdown pass earns 6 points, and Tom made 3 successful touchdown passes. I'll multiply the number of touchdowns by the points per touchdown to get the total points from touchdowns.Then, I'll calculate the points from the kicker's field goals. Each field goal is worth 3 points, and there were 2 successful field goals. I'll multiply the number of field goals by the points per field goal to find the total points from field goals.Finally, I'll add the points from touchdowns and field goals together to find the total points scored by the team.

answer：First, determine the total number of tickets Sarah needs to buy. Since she attends each of the 10 home games and buys one ticket for herself and one for her friend, she needs 2 tickets per game. This results in 20 tickets in total.Next, calculate the cost of the tickets. Each ticket costs 15, so the total cost for tickets is 20 tickets multiplied by 15, which equals 300.Then, calculate the cost of the scarves. After each game, Sarah buys one scarf costing 8. With 10 games, she buys 10 scarves, totaling 80.Finally, add the total cost of tickets and scarves together to find the overall amount Sarah will spend. 300 for tickets plus 80 for scarves equals 380.

answer：First, I need to determine how many athletes are competing in each of the three sports: wheelchair basketball, para-swimming, and para-athletics.The total number of athletes is 120.One-third of the athletes will compete in wheelchair basketball. To find this number, I calculate one-third of 120:120 × (1/3) = 40 athletes.One-quarter of the athletes will compete in para-swimming. To find this number, I calculate one-quarter of 120:120 × (1/4) = 30 athletes.The remaining athletes will compete in para-athletics. To find this number, I subtract the number of athletes in wheelchair basketball and para-swimming from the total number of athletes:120 - 40 - 30 = 50 athletes.Therefore, 50 athletes will compete in para-athletics.

answer：First, I need to determine the number of training sessions the judoka attended each week. Since they trained 5 days a week and each session lasted 2 hours, the weekly training time is 5 multiplied by 2, which equals 10 hours per week.Next, I'll calculate the total training time over the 8 weeks. By multiplying the weekly training hours (10) by the number of weeks (8), I find that the judoka spent a total of 80 hours training.

answer：First, I need to determine the current number of trips made by the existing buses. There are 3 buses, each making 4 trips per day, so currently, there are 3 multiplied by 4, which equals 12 trips per day.Next, the council member wants each bus to make 6 trips per day. Increasing each bus's trips from 4 to 6 means each bus will add 2 more trips. With 3 buses, this results in an additional 3 multiplied by 2, which is 6 trips. Adding this to the current total, the new number of trips with the existing buses will be 12 plus 6, totaling 18 trips per day.Additionally, the council member plans to add 2 more buses to the fleet. Each of these new buses will also make 6 trips per day. Therefore, the trips contributed by the new buses will be 2 multiplied by 6, which equals 12 trips per day.Finally, to find the total number of trips made by all the buses after the improvements, I add the trips from the existing buses and the new buses together: 18 plus 12 equals 30 trips per day.

answer：Okay, so I have this problem about Alex, a young actor, who is preparing a historical play. He needs to model the timeline of events using a directed graph. Each vertex is an event, and each edge shows the chronological order. The first part is about finding the longest path in this DAG, which is a directed acyclic graph. The function f(G) should find this longest path and calculate its total duration. I need to prove that this is equivalent to solving an optimization problem and explain how it helps in determining the optimal sequence of scenes.Hmm, let me think. So, the longest path in a DAG is a classic problem. It's used in scheduling and critical path analysis. Since each event has a duration, the longest path would represent the sequence of events that take the most time. In the context of the play, this would mean the optimal sequence of scenes that follow the historical timeline accurately, without skipping any necessary events or causing overlaps.To find the longest path, I remember that you can topologically sort the graph and then relax the edges in that order. This ensures that when you process each node, all its predecessors have already been processed, so you can update the longest path correctly. This method is efficient and works in linear time relative to the number of nodes and edges.So, f(G) is essentially solving the problem of finding the critical path in project scheduling, which is an optimization problem where you want to minimize the project duration. But in this case, since we're dealing with durations of events, the longest path would give the minimal sequence that covers all necessary events without violating the timeline. Wait, actually, no. The longest path would give the sequence that takes the most time, which might not necessarily be the minimal. Maybe I need to clarify that.Wait, in project scheduling, the critical path is the longest path because it determines the minimum time to complete the project. So, in this case, the longest path in the DAG would represent the minimal time required to cover all events in order, right? Because if you have dependencies, you can't overlap them, so the longest path gives the minimal makespan. So, in this case, f(G) is equivalent to solving the critical path method, which is an optimization problem to find the minimal time to complete all tasks given dependencies.Therefore, applying this to the play, the longest path would give the sequence of scenes that respect the historical timeline and cover all necessary events in the minimal possible time, ensuring that the play is both accurate and efficient.Moving on to the second part. During rehearsal, Alex finds that two key events were misplaced, creating a cycle. So, the graph is no longer a DAG. The problem is to devise an algorithm to detect and correct this inconsistency by removing the minimum number of edges to restore the DAG structure. I need to discuss the computational complexity and how this adjustment maintains historical accuracy.First, detecting cycles in a directed graph can be done using depth-first search (DFS). If during DFS, we find a back edge, that indicates a cycle. But since the graph is almost a DAG except for a couple of edges, maybe we can use a more efficient method.But once a cycle is detected, how do we remove the minimum number of edges to make it a DAG? This sounds like the problem of finding a feedback arc set, which is the smallest set of edges whose removal makes the graph acyclic. However, feedback arc set is an NP-hard problem, which means it's computationally intensive for large graphs. But in this case, since only two events are misplaced, maybe the cycle is small, so the problem is manageable.Alternatively, since the cycle is formed by two events and another existing event, perhaps it's a triangle or a small cycle. So, maybe we can find the cycle, identify the edges involved, and remove the one that is least necessary or the one that is incorrect.But to formalize this, perhaps we can use a topological sorting approach. If the graph has a cycle, topological sort will fail. So, we can try to perform a topological sort, and if it fails, identify the cycle. Then, for the detected cycle, remove edges until the cycle is broken.But how to choose which edges to remove? We need to remove the minimum number of edges. Since the cycle is small, maybe just removing one edge would suffice. But which one?Alternatively, we can model this as finding the minimum feedback arc set for the detected cycle. For a cycle of length k, the minimum feedback arc set is 1, since removing any single edge from the cycle will break it.But in the problem, it's mentioned that two key events were inaccurately placed, creating a cycle with another event. So, perhaps the cycle is of length 3, involving the two misplaced events and one existing event.Therefore, to correct this, we can detect the cycle, identify the edges that form it, and remove one edge that is incorrect. Since we need to maintain historical accuracy, we should remove the edge that doesn't correspond to the actual historical timeline.But how do we determine which edge is incorrect? We need some additional information or constraints. Maybe the two misplaced events have a known correct order, so we can adjust the edges accordingly.Alternatively, we can use domain knowledge about the historical events to decide which edge should be removed. For example, if event A should come before event B, but the graph has an edge from B to A, we can remove that edge.In terms of computational complexity, detecting a cycle can be done in O(n + m) time, where n is the number of nodes and m is the number of edges. Once the cycle is detected, identifying the minimum feedback arc set for a small cycle is manageable, as it's a constant time operation if the cycle is small.However, in the general case, finding the minimum feedback arc set is NP-hard, but since the inconsistency is only due to two misplaced events, the cycle is likely small, making the problem tractable.So, the algorithm would be:1. Perform a topological sort on the graph. If it fails, a cycle exists.2. Identify the cycle(s). Since only two events are misplaced, focus on the cycle involving these two and another event.3. For the identified cycle, determine which edge is incorrect based on historical accuracy.4. Remove the incorrect edge(s) to break the cycle and restore the DAG.This adjustment ensures that the timeline remains accurate, as we're correcting the order of events based on historical facts.I think that covers both parts. The first part is about finding the longest path as an optimization problem, and the second is about cycle detection and correction with minimal edge removal.**Final Answer**1. The function ( f(G) ) finds the longest path in the DAG, which corresponds to the optimal sequence of scenes. This is equivalent to solving the critical path method in project scheduling, ensuring the play's timeline is accurate and efficient. The result is (boxed{text{the longest path in the DAG}}).2. To correct the inconsistency, detect the cycle using DFS, identify the incorrect edge, and remove it. This maintains historical accuracy with minimal disruption. The computational complexity is manageable for small cycles, ensuring the DAG is restored. The solution is (boxed{text{removing the minimum number of edges to eliminate cycles}}).