rent-4-wheels makes a profit of $12 per day for each car rented, and loses $4 per day for each car that is not rented. How many cars should the company have on hand to maximize profit, subject to the following probability distribution:
anticipated number of rental requests
12 11 10 9 8 7 6
Probability
.09 .011 .22 .24 .12 .12 .10
This is what I did and I;m not sure if it's correct.
(12)(.09)+(4)(.91)=4.72
(11)(.11)+(4)(.89)=4.77
ect. i got that company should have ten cars to maximize profit
To determine the number of cars the company should have on hand to maximize profit, you need to calculate the expected profit for each possible number of cars and choose the number that yields the highest profit.
Let's walk through the calculations step by step using the given probability distribution:
1. For 12 cars:
Expected profit = (12 cars rented) * ($12 profit per car) * (0.09 probability) + (0 cars rented) * (-$4 loss per car) * (0.91 probability)
= $12 * 12 * 0.09 - $4 * 0.91
= $12.96 - $3.64
= $9.32
2. Repeat this calculation for each number of cars from 11 to 6.
Expected profit for 11 cars = $12 * 11 * 0.011 - $4 * 0.989
Expected profit for 10 cars = $12 * 10 * 0.22 - $4 * 0.78
Expected profit for 9 cars = $12 * 9 * 0.24 - $4 * 0.76
Expected profit for 8 cars = $12 * 8 * 0.12 - $4 * 0.88
Expected profit for 7 cars = $12 * 7 * 0.12 - $4 * 0.88
Expected profit for 6 cars = $12 * 6 * 0.10 - $4 * 0.90
3. Compare the expected profits for each number of cars and choose the maximum.
From the calculations, you can determine that the expected profits are as follows:
Expected profit for 12 cars = $9.32
Expected profit for 11 cars = $5.19
Expected profit for 10 cars = $22.20
Expected profit for 9 cars = $19.32
Expected profit for 8 cars = $8.96
Expected profit for 7 cars = $6.16
Expected profit for 6 cars = $4.80
Thus, the company should have 10 cars on hand to maximize its profit, as it yields the highest expected profit of $22.20.