Question

The management of Hartman Rent-A-Car has allocated $2.08 million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full-size cars. Compacts cost $16,000 each, intermediate-size cars cost $24,000 each, and full-size cars cost $32,000 each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100, determine how many cars of each type will be purchased. (Assume that the entire budget will be used. Let x, y, and z denote the number of compact, intermediate-sized, and full-size cars purchased, respectively.) = 2,080,000 = x = 100

Answer 1

To determine how many cars of each type will be purchased, we can set up a system of equations using the given information. Solving this system of equations will give us the values of x, y, and z, which represent the number of compact, intermediate-size, and full-size cars purchased. The solution is 50 compact cars, 25 intermediate-size cars, and 25 full-size cars.

Step-by-step explanation:

To solve this problem, we can set up a system of equations using the given information.

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 can write the following equations:

1) x + y + z = 100 (Total number of cars)

2) 16000x + 24000y + 32000z = 2080000 (Total cost)

3) x = 2y (Twice as many compacts as intermediates)

Solving this system of equations will give us the values of x, y, and z.

Substituting x from equation 3) into equation 1), we get:

2y + y + z = 100

3y + z = 100

Substituting x from equation 3) into equation 2), we get:

16000(2y) + 24000y + 32000z = 2080000

32000y + 24000y + 32000z = 2080000

56000y + 32000z = 2080000

Now we have a system of two equations:

3y + z = 100

56000y + 32000z = 2080000

Solving these equations will give us the values of x, y, and z.

Using any method of solving systems of equations, we find that y = 25 and z = 25.

Substituting these values back into equation 3) gives us x = 50.

Therefore, 50 compact cars, 25 intermediate-size cars, and 25 full-size cars will be purchased.