Questions: The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 60 students, requires 3 chaperones, and costs 1,200 to rent. Each van can transport 10 students, requires 1 chaperone, and costs 90 to rent. Since there are 540 students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 540 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs? The officers should rent buses and vans to minimize the transportation costs. The minimal transportation costs are .

The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 60 students, requires 3 chaperones, and costs 1,200 to rent. Each van can transport 10 students, requires 1 chaperone, and costs 90 to rent. Since there are 540 students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 540 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs? The officers should rent buses and vans to minimize the transportation costs. The minimal transportation costs are .
Transcript text: The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 60 students, requires 3 chaperones, and costs $\$ 1,200$ to rent. Each van can transport 10 students, requires 1 chaperone, and costs $\$ 90$ to rent. Since there are 540 students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 540 students. Since only 36 parents have volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs? The officers should rent $\square$ buses and $\square$ vans to minimize the transportation costs. The minimal transportation costs are \$ $\square$ .
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Solution

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Solution Steps

To solve this problem, we need to set up a system of linear inequalities based on the constraints given (number of students, number of chaperones, and cost). We will then use these inequalities to find the combination of buses and vans that minimizes the total cost. This is a linear programming problem, which can be solved using Python's optimization libraries.

  1. Define variables for the number of buses and vans.
  2. Set up inequalities for the number of students and chaperones.
  3. Define the cost function to minimize.
  4. Use a linear programming solver to find the optimal solution.
Step 1: Define Variables and Constraints

Let \( x \) be the number of buses rented and \( y \) be the number of vans rented. The constraints based on the problem are:

  1. Each bus can transport 60 students, and each van can transport 10 students. Therefore, the total number of students transported must satisfy: \[ 60x + 10y \geq 540 \]
  2. Each bus requires 3 chaperones, and each van requires 1 chaperone. Thus, the total number of chaperones used must satisfy: \[ 3x + y \leq 36 \]
Step 2: Cost Function

The total cost \( C \) for renting \( x \) buses and \( y \) vans is given by: \[ C = 1200x + 90y \] Our goal is to minimize \( C \).

Step 3: Solve the Linear Programming Problem

From the solution obtained, we find:

  • The optimal number of buses rented is approximately \( x = 6 \).
  • The optimal number of vans rented is approximately \( y = 18 \).
  • The minimal transportation cost is \( C = 8820 \).

Final Answer

The officers should rent \( \boxed{6} \) buses and \( \boxed{18} \) vans to minimize the transportation costs. The minimal transportation costs are \( \boxed{8820} \).

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