Solve the problem.
-Suppose the supply function of a certain item is given by $S(x)=2 x+7$ and the demand function is $D(x)=27-x^{2 / 3}$. Find the consumer's surplus.
A) $\frac{32}{5}$
B) 32
C) $\frac{64}{5}$
D) 64

Question

Quizplus · Accepted Answer

The consumer's surplus is calculated as the area between the demand curve and the price level at equilibrium quantity. First, find the equilibrium by setting $S(x) = D(x)$: $2x + 7 = 27 - x^{2/3}$. Solving gives $x = 8$. The equilibrium price is $S(8) = 2(8) + 7 = 23$. The consumer's surplus is $\int_0^8 (27 - x^{2/3}) \, dx - 8 \times 23$. Calculating this integral and subtracting the rectangle area gives $\frac{64}{5}$.

Solve the Problem S(x)=2x+7S(x)=2 x+7S(x)=2x+7 And the Demand Function Is D(x)=27−x2/3D(x)=27-x^{2 / 3}D(x)=27−x2/3

Solve the Problem $S(x)=2 x+7$ And the Demand Function Is $D(x)=27-x^{2 / 3}$