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