Final answer:
To find the rate that maximizes income for a car rental agency, we need to calculate the marginal revenue and marginal cost for each rate increment. By comparing the total revenue and total cost at each rate, we can identify the optimal rate and the corresponding maximum income.
Step-by-step explanation:
To determine the rate at which the cars should be rented to maximize income, we need to find the point at which the marginal revenue equals the marginal cost. The marginal revenue is the additional revenue earned from renting one more car, while the marginal cost is the additional cost incurred. In this case, we can calculate the marginal revenue and marginal cost for each rate increment to find the optimal rate.
To calculate the marginal revenue, we need to consider the number of cars rented at each rate increment. For every $1 increase in rate, 5 fewer cars are rented. Starting with the given information that 190 cars are rented per day at a rate of $40 per day, we can calculate the number of cars rented at different rate increments. For example, at a rate of $41 per day, 185 cars are rented (190 - 5). Similarly, at a rate of $42 per day, 180 cars are rented, and so on.
Next, we can calculate the total revenue and total cost for each rate increment. The total revenue is the product of the number of cars rented and the rate, while the total cost is the fixed cost plus the variable cost (number of cars rented times the rate). For example, at a rate of $41 per day, the total revenue is (185 cars * $41) = $7,585 and the total cost is ($7,585 + $10,000) = $17,585 ($10,000 for fixed cost and $7,585 for variable cost). Similarly, we can calculate total revenue and total cost at each rate increment.
By comparing the total revenue and total cost at each rate increment, we can identify the rate that maximizes the income. This is the rate at which the difference between total revenue and total cost is the highest. The corresponding income is the difference between the total revenue and the total cost at the optimal rate.
In summary, to find the rate at which the cars should be rented to produce the maximum income, we calculate the marginal revenue and marginal cost for each rate increment. Then, we compare the total revenue and total cost at each rate to identify the optimal rate. The maximum income corresponds to the difference between the total revenue and total cost at the optimal rate.