Given: ( $x$ is number of items) Demand function: $d(x)=\frac{1568}{\sqrt{x}}$ Supply function: $s(x)=2 \sqrt{x}$ Find the equilibrium quantity: items Find the consumers surplus at the equilibrium quantity: $\$$

Step 1 ：Given that the demand function is \(d(x)=\frac{1568}{\sqrt{x}}\) and the supply function is \(s(x)=2 \sqrt{x}\), we need to find the equilibrium quantity and the consumer's surplus at the equilibrium quantity.

Step 2 ：The equilibrium quantity is found when the demand function equals the supply function. So, we need to solve the equation \(d(x) = s(x)\) for \(x\).

Step 3 ：Solving the equation gives us \(x = 784\). This is the equilibrium quantity.

Step 4 ：To find the consumer's surplus at the equilibrium quantity, we need to calculate the area between the demand curve and the price (which is the supply curve evaluated at the equilibrium quantity) from 0 to the equilibrium quantity.

Step 5 ：The consumer surplus is given by the integral \(\int_0^{x_e} (d(x) - p) dx\), where \(x_e\) is the equilibrium quantity and \(p\) is the price at the equilibrium quantity.

Step 6 ：Evaluating the integral gives us a consumer surplus of \$43904.

Step 7 ：Final Answer: The equilibrium quantity is \(\boxed{784}\) items and the consumer's surplus at the equilibrium quantity is \(\boxed{\$43904}\).