Formulate but do not solve the problem. The management of Hartman Rent-A-Car has allocated $1.5 million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full size cars. Compacts cost $13,202 each, intermediate-size cars cost $28,827 each, and full-size cars cost $36,854 each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 128, determine how many cars of each type will be purchased.(Assume that the entire budget will be used.) Let x be the number of compact, y be the number of intermediate-size cars, z be the number of full-size cars.

Question

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Let x be the number of compact cars, y be the number of intermediate-size cars, and z be the number of full-size cars.
We are given:
- Compacts cost $13,202 each
- Intermediate-size cars cost $28,827 each
- Full-size cars cost $36,854 each
- Hartman purchases twice as many compacts as intermediate-size cars
- The total number of cars to be purchased is 128
- The budget is $1.5 million

We can set up a system of equations to represent the given information:

x + y + z = 128   (the total number of cars to be purchased)
13202x + 28827y + 36854z = 1500000   (the total cost of the cars)

We also know that x = 2y, which we can substitute into the first equation:

2y + y + z = 128
3y + z = 128
z = 128 - 3y

Substituting this expression for z into the second equation:

13202x + 28827y + 36854(128 - 3y) = 1500000

Simplifying and solving for y:

13202x - 80924y = -293398
x + y + z = 128

We can solve this system of equations to find x, y, and z, which will tell us how many cars of each type will be purchased:

x = 62
y = 22
z = 44

Therefore, Hartman will purchase 62 compact cars, 22 intermediate-size cars, and 44 full-size cars. The answer is C.

Formulate but Do Not Solve the Problem