Question

The annual revenue and cost functions for a manufacturer of zip drives are approximately R(x)=520x-0.02x² and C(x) = 160x+100,000, where x denotes the number of drives made. What is the maximum annual profit? A. $1,620,000 B. $1,720,000 C. $1,520,000 D. $1,820,000

Answer 1

To determine the maximum annual profit, we need to calculate the profit at different levels of output using the given revenue and cost functions.

Step-by-step explanation:

Calculating Total Revenue and Total Cost for Different Levels of Output:

To calculate the total revenue at different levels of output, we can use the revenue function R(x) = 520x - 0.02x^2. Substitute the values of x (number of drives made) into the function to find the total revenue.

To calculate the total cost at different levels of output, we can use the cost function C(x) = 160x + 100,000. Substitute the values of x (number of drives made) into the function to find the total cost.

Calculating Profit:

Once we have the total revenue and total cost, we can calculate the profit by subtracting the total cost from the total revenue. The maximum annual profit will occur when the profit is maximized.

Answer:

The maximum annual profit can be determined by calculating the profit at different levels of output and finding the highest value. To determine the answer, we need to calculate the profit for different levels of output using the given revenue and cost functions.

Answer 2

The maximum annual profit for the manufacturer of zip drives is calculated by finding the profit function and its derivative, setting the derivative to zero to find the critical point, and then using this to calculate the profit. The maximum profit is determined to be $1,520,000.

Step-by-step explanation:

The student has asked for the calculation of the maximum annual profit for a manufacturer of zip drives with given revenue and cost functions, R(x) = 520x - 0.02x² and C(x) = 160x + 100,000. To determine this, we must first find the profit function by subtracting the cost function from the revenue function, P(x) = R(x) - C(x). Then, we need to find the derivative of this profit function to determine the number of drives that would yield the maximum profit. We set this derivative equal to zero to find the critical points and then check these points to find the maximum profit. The options provided are hinting towards very large profits, so we should expect x to be large as well when maximizing profit.

First, calculate the profit function:
P(x) = R(x) - C(x)
P(x) = (520x - 0.02x²) - (160x + 100,000)
P(x) = 360x - 0.02x² - 100,000.

Next, find the derivative of the profit function:
P'(x) = 360 - 0.04x.

To find the maximum profit, set the derivative equal to zero and solve for x:
0 = 360 - 0.04x
0.04x = 360
x = 9000.

Now, use this value of x to calculate the maximum profit:
P(9000) = 360(9000) - 0.02(9000)² - 100,000
P(9000) = 3,240,000 - 1,620,000 - 100,000
P(9000) = $1,520,000.

The maximum annual profit is therefore $1,520,000, which corresponds to option C.