Question

a manufacturer has determined that the revenue from the sale of \\(x\\) cordless telephones is given by \\(r(x) = 89x - 0.01x^2\\) dollars. the cost of producing \\(x\\) telephones is \\(c(x) = 10,800 + 19x\\) dollars. find the marginal profit \\(p(300)\\).

Explanation:

Formulate the profit function

$$ P(x) = R(x) - C(x) = (89x - 0.01x^2) - (10,800 + 19x) = -0.01x^2 + 70x - 10,800 $$

Differentiate to find marginal profit

$$ P'(x) = \frac{d}{dx}(-0.01x^2 + 70x - 10,800) = -0.02x + 70 $$

Evaluate at x = 300

$$ P'(300) = -0.02(300) + 70 = -6 + 70 = 64 $$

Answer:

A manufacturer has determined that the revenue from the sale of \(x\) cordless telephones is given by \(R(x) = 89x - 0.01x^2\) dollars. The cost of producing \(x\) telephones is \(C(x) = 10,800 + 19x\) dollars. Find the marginal profit \(P'(300)\).

\(P'(300) =\) <blank>64</blank> dollars