Find the consumers' surplus if the demand function for a particular beverage is given by $D(q)=\frac{3000}{(3 q+7)^{2}}$ and if the supply and demand are in equilibrium at $q=9$.

Step 1 ：Given the demand function for a particular beverage is \(D(q)=\frac{3000}{(3 q+7)^{2}}\) and the supply and demand are in equilibrium at \(q=9\).

Step 2 ：We need to find the consumer surplus, which is the area between the demand curve and the price level up to the quantity demanded.

Step 3 ：This can be found by integrating the demand function from 0 to the equilibrium quantity, and then subtracting the total amount of money consumers would have been willing to pay at the equilibrium price.

Step 4 ：The equilibrium price can be found by substituting the equilibrium quantity into the demand function, which gives us \(P = \frac{750}{289}\).

Step 5 ：Substituting these values into the formula for consumer surplus gives us a consumer surplus of \(\frac{182250}{2023}\).

Step 6 ：Final Answer: The consumers' surplus is \(\boxed{\frac{182250}{2023}}\).