Question

An office supply company sells x heavy-duty paper shredders per year at Sp per shredder. The price demand equation and the cost function for these shredders are given by:

p = 400 -0.4x and C(x) = 2,000+160x.
What is the maximum yearly profit?

Answer 1

To find the maximum yearly profit, we need to subtract the cost from the revenue. The profit function is P(x) = (400 - 0.4x)x - (2,000 + 160x). Taking the derivative of the profit function with respect to x, setting it equal to 0, and solving for x gives us the value that maximizes the profit. Substituting this value back into the profit function gives us the maximum yearly profit of $108,400.

Step-by-step explanation:

To find the maximum yearly profit, we need to subtract the cost from the revenue. The revenue is given by the price demand equation, which is p = 400 - 0.4x, and the cost is given by the cost function, which is C(x) = 2,000 + 160x. The profit function is given by P(x) = (400 - 0.4x)x - (2,000 + 160x). To find the maximum profit, we need to find the value of x that maximizes the profit function.

To do this, we can take the derivative of the profit function with respect to x, set it equal to 0, and solve for x. Once we have the value of x, we can substitute it back into the profit function to find the maximum profit.

Taking the derivative of the profit function P(x), we get P'(x) = -0.4x + 400 - 160. Setting this equal to 0, we have -0.4x + 240 = 0. Solving for x, we find x = 600.

Substituting x = 600 back into the profit function P(x), we get P(600) = (400 - 0.4(600))(600) - (2,000 + 160(600)). Calculating this, we find that the maximum yearly profit is $108,400.

Answer 2

To find the maximum yearly profit, calculate the revenue and cost functions and subtract the cost from the revenue. Find the x-value at which the derivative of the profit function is zero, and calculate the profit at that point.

Step-by-step explanation:

To find the maximum yearly profit, we need to calculate the revenue and cost functions. The revenue function can be found by multiplying the price per shredder (p) by the number of shredders sold per year (x). So the revenue function is R(x) = px = (400 - 0.4x)x.

The cost function is given as C(x) = 2,000 + 160x.

To find the profit function, subtract the cost function from the revenue function: Profit(x) = R(x) - C(x).

To find the maximum yearly profit, we need to find the x-value at which the derivative of the profit function is equal to zero. Differentiating the profit function with respect to x, we get d(Profit(x))/dx = 400 - 0.8x - 160 = 0. Solving for x, we find x = 200. plugging this value back into the profit function gives Profit(200) = (400 - 0.4(200))(200) - (2,000 + 160(200)) = $40,000.