Question 9
Given: ( $x$ is number of items)
Demand function: $d(x)=\frac{2304}{\sqrt{x}}$
Supply function: $s(x)=4 \sqrt{x}$
Find the equilibrium quantity: 576
items
Find the consumers surplus at the equilibrium quantity: $\$$
Question Help: $\square$ Video $1 \square$ Video 2

Step 1 ：Given the demand function \(d(x)=\frac{2304}{\sqrt{x}}\) and the supply function \(s(x)=4 \sqrt{x}\), we need to find the equilibrium quantity. This is the quantity at which the demand and supply functions intersect. Therefore, we set the demand function equal to the supply function and solve for x.

Step 2 ：Setting \(d(x) = s(x)\), we get \(\frac{2304}{\sqrt{x}} = 4 \sqrt{x}\). Solving this equation, we find that the equilibrium quantity is 576 items.

Step 3 ：To find the consumer's surplus at the equilibrium quantity, we calculate the area between the demand curve and the price level up to the equilibrium quantity. The consumer surplus is given by the integral of the demand function from 0 to the equilibrium quantity minus the total cost which is the price times the equilibrium quantity.

Step 4 ：Given that the price at equilibrium is 96, the consumer surplus is given by \(\int_{0}^{576} \frac{2304}{\sqrt{x}} dx - 96*576\).

Step 5 ：Calculating the above expression, we find that the consumer's surplus at the equilibrium quantity is \(\boxed{55296}\).