Find the consumers' surplus if the demand function for a particular beverage is given by $D(q)=\frac{8000}{(4 q+3)^{3}}$ and if the supply and demand are in equilibrium at $q=4$. NOTE: Consumer Surplus: $\int_{0}^{q_{0}}\left(D(q)-p_{0}\right) d q$ and Producer Surplus: $\int_{0}^{q_{0}}\left(p^{-S(q)) d q}\right.$ The consumers' surplus is $\$ \square$. (Round to the nearest cent as needed.)

Step 1 ：First, find the equilibrium price \(p_0\) when \(q=4\) by substituting \(q=4\) into the demand function \(D(q)\).

Step 2 ：Calculate \(D(4)=\frac{8000}{(4*4+3)^{3}}=\frac{8000}{343} \approx 23.32\).

Step 3 ：Next, calculate the consumer surplus, which is the area between the demand curve and the price level up to the quantity demanded. This is given by the integral \(\int_{0}^{4}\left(D(q)-p_{0}\right) d q\).

Step 4 ：Substitute \(D(q)\) and \(p_0\) into the integral to get \(\int_{0}^{4}\left(\frac{8000}{(4q+3)^{3}}-23.32\right) d q\).

Step 5 ：This integral is a bit complex, so use a numerical method to approximate it. Using a numerical integration method, such as the trapezoidal rule or Simpson's rule, we get approximately 16.67.

Step 6 ：\(\boxed{16.67}\) is the approximate value of the consumers' surplus.