Find the producers' surplus if the supply function for pork bellies is given by the following.
\[
S(q)=q^{5 / 2}+3 q^{3 / 2}+54
\]

Assume supply and demand are in equilibrium at $q=4$.
NOTE: Consumer Surplus: $\int_{0}^{q_{0}}\left(D(q)-p_{0}\right) d q$ and Producer Surplus: $\int_{0}^{q_{0}}\left(p_{0}-S(q)\right) d q$

The producers' surplus is $\$ \square$.
(Type an integer or decimal rounded to the nearest hundredth as needed.)

Step 1 ：First, we need to find the equilibrium price by substitifying the equilibrium quantity into the supply function. The supply function is given by \(S(q) = q^{5/2} + 3q^{3/2} + 54\). Substituting \(q = 4\) into the supply function, we get \(p_0 = 110\).

Step 2 ：Next, we calculate the producer's surplus. The producer's surplus is given by the integral of the difference between the equilibrium price and the supply function, from 0 to the equilibrium quantity. Therefore, we calculate the integral of \(p_0 - S(q)\) from 0 to 4.

Step 3 ：By calculating the integral, we find that the producer's surplus is approximately 149.03.

Step 4 ：Final Answer: The producers' surplus is \(\boxed{149.03}\).