15.07.2022 - 05:13

# A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $10,000, and the variable cost for producing x pagers/week in dollars is represented by the function V Question: A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is$10,000, and the variable cost for producing x pagers/week in dollars is represented by the function V(x).

{eq}V(x) = 0.000001x^3 – 0.01x^2 + 50x {/eq}

The company realizes a revenue in dollars from the sale of x pagers/week represented by the function R(x).

{eq}R(x) = -0.02x^2 + 150x space (0 \leq x \leq 7500) {/eq}

(a) Find the total cost function C.

(b) Find the total profit function P.

(c) What is the profit for the company if 1,700 units are produced and sold each week?

(a) To find the total cost function C, we add the weekly fixed cost of 10,000 to the variable cost function V(x): {eq}begin{aligned} C(x) &= 10,000 + V(x) \ &= 10,000 + (0.000001x^3 - 0.01x^2 + 50x) end{aligned} {/eq} (b) To find the total profit function P, we subtract the total cost function C from the revenue function R: {eq}begin{aligned} P(x) &= R(x) - C(x) \ &= (-0.02x^2 + 150x) - (10,000 + 0.000001x^3 - 0.01x^2 + 50x) \ &= -0.000001x^3 + 0.01x^2 + 100x - 10,000 end{aligned} {/eq} (c) To find the profit for the company if 1,700 units are produced and sold each week, we substitute x = 1,700 into the profit function: {eq}begin{aligned} P(1,700) &= -0.000001(1,700)^3 + 0.01(1,700)^2 + 100(1,700) - 10,000 \ &= 357,200 end{aligned} {/eq} Therefore, the profit for the company if 1,700 units are produced and sold each week is357,200.