Find the producers' surplus if the supply function for pork bellies is given by the following. \[ S(q)=q^{5 / 2}+2 q^{3 / 2}+52 \] Assume supply and demand are in equilibrium at $q=25$. The producers' surplus is $\$$. (Type an integer or decimal rounded to the nearest hundredth as needed.)

Step 1 ：The supply function for pork bellies is given by \(S(q)=q^{5 / 2}+2 q^{3 / 2}+52\).

Step 2 ：The equilibrium quantity is given as \(q=25\).

Step 3 ：The equilibrium price can be found by substituting \(q=25\) into the supply function, which gives us \(3427.00\).

Step 4 ：The producer surplus can then be calculated as the integral of the supply function from 0 to the equilibrium quantity, subtracted from the product of the equilibrium price and quantity.

Step 5 ：By calculating, we find that the producer surplus is \(59553.57\).

Step 6 ：Final Answer: The producers' surplus is \(\boxed{59553.57}\).