Question

Suppose the marginal cost curve is given by the function q^2 + 5q + 30. If the demand curve is the same as before, what will be market price and market quantity?

Answer 1

Suppose the demand curve is given by $P=120-.5Q$

$TR=P\times Q\\=(120-0.5Q)Q\\=120Q-0.5Q^2\\MR=120-Q$

$MC=Q^2+5Q+30$

$MR=MC\\120-Q=Q^2+5Q+30\\Q^2+5Q+Q+30-120=0\\Q^2+6Q-90=0$

To find Q we use the quadratic formula

$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$\frac{-6\pm\sqrt{6^2-(4\times-90)}}{2}$

$-6\pm \frac{\sqrt{36+360}}{2}$

$\frac{-6\pm\sqrt{36+360}}{2}$

$\frac{-6\pm\sqrt{396}}{2}$

$\frac{-6+\sqrt{396}}{2}\space or\space \frac{-6-\sqrt{396}}{2}$

$=6.949874371\space or\space -12.94987437$

therefore $Q\approx7$

$P=120-.5(7)=116.5$