Problema Solution
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 4 chaperones, and costs $1,200 to rent. Each van can transport 8 students, requires 1 chaperone, and costs $90 to rent. Since there are 560 students in the senior class that may be eligible to go on the trip, the officers must plan to accommodate at least 560 students. Since only 60 parents have volunteered to serve as chaperones, the officers must plan to use at most 60 chaperones. How many vehicles of each type should the officers rent in order to minimize the transportation costs? What are the minimal transportation costs?
Answer provided by our tutors
Let
x = number of buses
y = number of vans
the constrains are:
40x + 8y >= 560 (the officers must plan to accommodate at least 560 students)
4x + y <= 60 (only 60 parents have volunteered to serve as chaperones)
x >= 0
y >= 0
click here to see the graph of the above system of inequalities:
the critical points are:
(14, 0) (we get this point by solving the system of equations: 40x + 8y = 560, y = 0)
(15, 0) (we get this point by solving the system of equations: 4x + y = 60, y = 0)
(10, 20) (we get this point by solving the system of equations: 40x + 8y = 560, 4x + y = 60)
the objective function is F(x, y) = 1200x + 90y we need to find the minimum
F(14, 0) = 1200*14 + 90*0 = $16,800
F(15, 0) = 1200*15 + 90*0 = $18,000
F(10, 20) = 1200*10 + 90*20 = $13,800
in order to minimize the transportation costs the officers should rent 10 buses and 20 vans.
the minimal transportation costs are $13,800.