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