We are given the total cost function $C(x) = -2350 + 80x - 0.04x^2$ and the price function $P(x) = 2x - 35$. We need to find: (a) The marginal profit function. (b) The level of production that maximizes profit, rounded to the nearest whole number. (c) The maximum annual profit, rounded to the nearest whole number.

1. Problem Description

We are given the total cost function $C(x) = -2350 + 80x - 0.04x^2$ and the price function $P(x) = 2x - 35$. We need to find:

(a) The marginal profit function.

(b) The level of production that maximizes profit, rounded to the nearest whole number.

(c) The maximum annual profit, rounded to the nearest whole number.

2. Solution Steps

(a) Find the marginal profit function.

First, we need to find the revenue function, $R(x)$. Revenue is the price per item times the number of items sold, so $R(x) = xP(x) = x(2x - 35) = 2x^2 - 35x$.

Next, we find the profit function $Pr(x)$. Profit is revenue minus cost, so

$Pr(x) = R(x) - C(x) = (2x^2 - 35x) - (-2350 + 80x - 0.04x^2) = 2x^2 - 35x + 2350 - 80x + 0.04x^2 = 2.04x^2 - 115x + 2350$.

The marginal profit function is the derivative of the profit function with respect to $x$.

$Pr'(x) = \frac{d}{dx}(2.04x^2 - 115x + 2350) = 4.08x - 115$.

(b) Find the level of production that maximizes profit.

To maximize profit, we set the marginal profit function equal to zero and solve for $x$.

$Pr'(x) = 4.08x - 115 = 0$

$4.08x = 115$

$x = \frac{115}{4.08} \approx 28.186$.

Since we need to round the answer to the nearest whole number, the level of production that maximizes profit is $x = 28$.

(c) Calculate the maximum annual profit.

We substitute the value of $x$ found in part (b) into the profit function.

$Pr(28) = 2.04(28)^2 - 115(28) + 2350 = 2.04(784) - 3220 + 2350 = 1599.36 - 3220 + 2350 = 729.36$.

Rounding to the nearest whole number, the maximum annual profit is

7

2

9.

3. Final Answer

(a) The marginal profit function is $Pr'(x) = 4.08x - 115$.

(b) The level of production that maximizes profit is

2

8. (c) The maximum annual profit is 729.