Question $4.$ Integral Calculus and Welfare Analysis

Consider the market for a good with the following inverse demand and supply functions:

$P=220-\frac{4}{5}{Q}^2$

$P=76+\frac{1}{5}{Q}^2$

$(Inverse$ Demand$)$

$(Inverse$ Supply$)$

Tasks

Find the equilibrium quantity $(Q^{*})$ and price $(P^{*}).$

Graph the given inverse demand and supply functions.

The drawing does not need $tobe$ precise, but $it$ should clearly indicate where

both the demand and supply functions intersect the vertical axis. Additionally,

make sure the functions are sloping $in$ the correct direction.

Clearly label the equilibrium price $(P^{*})$ and quantity $(Q^{*}).$

Clearly label the equilibrium point and name $itE.$

Clearly shade and label the areas representing the consumer surplus $(CS)$ and

producer surplus $(PS).$

Calculate the consumer surplus.