Find the producers' surplus if the supply function for pork bellies is given by the following \[ S(q)=q^{5 / 2}+2 q^{3 / 2}+55 \] Assume supply and demand are in equilibrium at $q=25$. The producers' surplus is $\$ 59553.57$

Step 1 ：The supply function for pork bellies is given by \(S(q)=q^{5 / 2}+2 q^{3 / 2}+55\).

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

Step 3 ：The equilibrium price is given by \(S(25)\), which is \(3430.00000000000\) dollars.

Step 4 ：The producer surplus is the area above the supply curve but below the equilibrium price. To find this, we need to integrate the supply function from 0 to the equilibrium quantity and subtract this value from the product of the equilibrium price and quantity.

Step 5 ：By doing this, we find that the producer surplus is \(59553.5714285714\) dollars.

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