Solve the problem.
-Find the consumer's surplus if the demand function for an item is given by $D(x)=30-x^{2}$, assuming supply and demand are in equilibrium at $x=4$.
A) $\frac{128}{3}$
B) 128
C) 64
D) $\frac{64}{3}$

Question

Quizplus · Accepted Answer

The consumer's surplus is calculated as the area between the demand curve and the equilibrium price level from 0 to the equilibrium quantity. The equilibrium price is $D(4) = 30 - 4^2 = 14$. The consumer's surplus is $\int_{0}^{4} (30 - x^2) \, dx - 4 \times 14$. Evaluating the integral gives $\left[30x - \frac{x^3}{3}\right]_{0}^{4} = 120 - \frac{64}{3}$. Subtracting the equilibrium expenditure $4 \times 14 = 56$, the consumer's surplus is $\frac{128}{3}$.

Solve the Problem D(x)=30−x2D(x)=30-x^{2}D(x)=30−x2 , Assuming Supply and Demand Are in Equilibrium At

Solve the Problem $D(x)=30-x^{2}$ , Assuming Supply and Demand Are in Equilibrium At