A company estimates that its marginal profit for producing a product is given by $\mathrm { MP } ( q ) = 60 - 0.12 q$ where $\operatorname { MP } ( q )$ is in dollars per unit. Given that $P = - 500$ when $\mathrm { q } = 0$ find the total profit realized from selling 300 units of product.
A) $12,100
B) $7700
C) $6700
D) $4100

Question

Quizplus · Accepted Answer

The total profit, $P(q)$, from selling $q$ units is found by integrating the marginal profit, $\mathrm{MP}(q) = 60 - 0.12q$, with respect to $q$. Given $P(0) = -500$, we integrate and apply the initial condition to find the constant of integration. The integral of $60 - 0.12q$ with respect to $q$ is $60q - 0.06q^2 + C$. Using $P(0) = -500$, we find $C = -500$. Thus, $P(q) = 60q - 0.06q^2 - 500$. Substituting $q = 300$ gives $P(300) = 60(300) - 0.06(300)^2 - 500 = 18000 - 5400 - 500 = 12100$.

A Company Estimates That Its Marginal Profit for Producing a Product