Chapter 5.3, Problem 53E

In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method.

Transportation. The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport $40$ students, requires $3$ chaperones, and costs $\text{\$}1,200$ to rent. Each van can transport $8$ students, requires $1$ chaperone, and costs $\text{\$}100$ to rent. Since there are $400$ students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least $400$ students. Since only $36$ parents have volunteered to serve as chaperones, the officers must plan to sue 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?