Answer to Question #235623 in Calculus for john k

Question

The demand equation for a product is

Q= $10*(100-P)^1/2$

Calculate the consumers’ surplus under equilibrium which occurs at a market price of K84. 

Answer 1

$$Q=10(100-P)^{1/2}$$

$$P=100-0.01Q$$

$$P^*=84=>Q^*=10(100-84)^{1/2}=40$$

The consumer surplus is

$$\displaystyle\int_{0}^{Q^*}P(Q)dQ-P^*Q^*$$

$$=\displaystyle\int_{0}^{40}(100-0.01Q)dQ-84(40)$$

$$=[100Q-0.005Q^2]\begin{matrix} 40 \\ 0 \end{matrix}-3360$$

$$=4000-8-3360=632$$

The consumers’ surplus under equilibrium is $632.$