Question

A car rental agency in a major city has a total of 4300 cars that it rents from three locations: Metropolis Airport, downtown, and the smaller City Airport. Some weekly rental and return patterns are shown in the table (note that Airport means Metropolis Airport).

Rented from
Returned to AP DT CA
Airport (AP) 90% 10% 10%
Downtown (DT) 5% 80% 5%

At the beginning of a week, how many cars should be at each location so that the same number of cars will be there at the end of the week (and hence at the start of the next week)?
AP
Correct: Your answer is correct.
cars
DT
Incorrect: Your answer is incorrect.
cars
CA
Incorrect: Your answer is incorrect.
cars

Answer 1

Explanation:

To determine the number of cars that should be at each location at the beginning of the week, we can use the rental and return patterns provided in the table.

Let's denote the number of cars at each location at the beginning of the week as follows:

AP: Number of cars at Metropolis Airport

DT: Number of cars at downtown

CA: Number of cars at City Airport

Based on the rental and return patterns, we can calculate the number of cars at each location at the end of the week (and hence at the start of the next week) using the following equations:

AP = 0.9(AP) + 0.05(DT) + 0.1(CA)

DT = 0.1(AP) + 0.8(DT) + 0.05(CA)

CA = 0.1(AP) + 0.05(DT) + 0.9(CA)

Since we want the same number of cars at each location at the end of the week, we can set the equations equal to each other:

AP = DT = CA

Now, we can solve these equations to find the number of cars at each location.

Let's denote the number of cars at each location as N:

AP = N

DT = N

CA = N

Substituting these values into the equations, we get:

N = 0.9(N) + 0.05(N) + 0.1(N)

N = 1.05(N)

Simplifying the equation, we have:

N = N/1.05

To solve for N, we can multiply both sides of the equation by 1.05:

1.05(N) = N

1.05N - N = 0

0.05N = 0

N = 0/0.05

N = 0

Therefore, the number of cars at each location at the beginning of the week should be 0.

So, the correct answer is:

AP: 0 cars

DT: 0 cars

CA: 0 cars