Question

A company has found that its supply function is equal to the square of the price of their product, all divided by three. The market for its products is also related to price. demand is a basic 920 units minus thirty times price, minus a quarter of the square of the price. What is the ideal price of their product, to make sure that there is neither a surplus nor a shortage? How many units will they sell at this price?

Answer 1

Given that the supply function for the company is equal to the square of the price of their product, all divided by three,

i.e. $Q_s=\frac{p^2}{3}$

and the demand is a basic 920 units minus thirty times price, minus a quarter of the square of the price,

i.e. $Q_d=920-30p- \frac{1}{4} p^2$

The ideal price of their product, to make sure that there is neither a surplus nor a shortage is the equilibrium price which is the price when supply equals demand.

i.e.
$Q_s=Q_d \\ \\ \Rightarrow\frac{p^2}{3}=920-30p- \frac{1}{4} p^2 \\ \\ \Rightarrow \frac{7}{12} p^2+30p-920=0 \\ \\ \Rightarrow p= \frac{-30\pm\sqrt{30^2-4(\frac{7}{12})(-920)}}{2(\frac{7}{12})} \\ \\ =\frac{-30\pm\sqrt{900+\frac{6440}{3}}}{\frac{7}{6}}=\frac{-30\pm\sqrt{\frac{9140}{3}}}{1.1667} \\ \\ =\frac{-30\pm55.1966}{1.1667}= \frac{25.1966}{1.1667} =21.60$

Therefore, the ideal price of their product, to make sure that there is neither a surplus nor a shortage is $21.60

The number of units they will sell at this price is given by

$Q= \frac{(21.60)^2}{3} \\ \\ = \frac{466.43}{3} \approx155\ units$