Q. 22 - Precalculus Enhanced with Graphing Utilities

Question

Spring Break The student activities department of a community college plans to rent buses and vans for a spring-break trip. Each bus has 40 regular seats and 1 handicapped seat; each van has 8 regular seats and 3 handicapped seats. The rental cost is $\backslash (350$ for each van and $\backslash )975$ for each bus. If 320 regular and 36 handicapped seats are required for the trip, how many vehicles of each type should be rented to minimize cost? 

Step-by-Step Solution

Answer

The minimum cost will be applicable if the number of vans bought is 10 and number of buses bought is 6 . 

Step 1. Given information

Let the number of buses and vans rented be x and y respectively.
The capacity of a bus is 40 regular seats and 1 handicapped seats, and each van has 8 regular seats and 3 handicapped seats. If a total of 320 regular seats and 36 handicapped seats are required for the trip, the conditions can be written in terms of x and y. 

$$\begin{gathered} 40x+80y\ge 320 \\ x+3y\ge 36 \end{gathered}$$

Step 2. The number of vans and buses cannot be less than zero; therefore the following inequalities are also required to be satisfied.

$$\begin{gathered} x\ge 0 \\ y\ge 0 \end{gathered}$$

The rental cost for each van is $\$350$ and the rental cost for each bus is $\$975$. The cost function can be determined as follows:

$C=350x+975y$

The region where we can evaluate the minimum cost function $C=350x+975y$ can be obtained by graphing the inequalities.

Step 3. Finding the value of C

We need to determine the value of the expression $C=350x+975y$ at each vertex in order to determine the minimum of the expression. We have given below the table of the values calculated at each of the vertex.

Vertex	Value of $C=350x+975y$
$(6,10)$	$C=350\left(6\right)+975\left(10\right)=11850$

Therefore, the minimum cost will be applicable if the number of vans bought is 10 and number of buses bought is 6 .