Question

The revenue (in dollars) for selling q items is given by R ( q ) = 348 q − 2 q 2 and the costs (in dollars) of producing q items is given by C ( q ) = 100 + 60 q . Find the quantity that gives the maximum profit.

Answer 1

The quantity of 72 that gives the maximum profit of 10268

Explanation:

The profit function p(q) is given by the difference between the revenue and the cost function,

                   P(q) = R(q) - C(q)

The revenue (in dollars) for selling q items is given by  $R(q)=348 q-2 q^{2}$

The costs (in dollars) of producing q items is given by C(q)= 100 + 60q

             $P(q)=348 q-2 q^{2}-(100+60 q)$

             $=348 q-2 q^{2}-100-60 q$

             $=-2 q^{2}+288 q-100$

The above profit function is a downward opening parabola. Its maximum value occurs,

           $\text { At } x=-\frac{b}{2 a}=-\frac{288}{2(-2)}=72$

Maximum value, $p(72)=\left(-2 \times 72^{2}\right)+(288 \times 72)-100=-10368+20736-100=10268$