To solve for the producer's surplus in this situation, we follow a series of steps that involve finding the equilibrium price, the area under the supply curve, and, finally, the producer's surplus. Here is the detailed step-by-step solution:
1. Find the Supply Function [tex]\( S(q) \)[/tex] at the Equilibrium Quantity [tex]\( q = 9 \)[/tex]:
The supply function is given by:
[tex]\[
S(q) = q^{5/2} + 3q^{3/2} + 52
\][/tex]
Plugging [tex]\( q = 9 \)[/tex] into the supply function to find the equilibrium price:
[tex]\[
S(9) = 9^{5/2} + 3 \cdot 9^{3/2} + 52
\][/tex]
Calculate each term separately:
[tex]\[
9^{5/2} = (9)^{2.5} = 243
\][/tex]
[tex]\[
3 \cdot 9^{3/2} = 3 \cdot (9)^{1.5} = 3 \cdot 27 = 81
\][/tex]
[tex]\[
52 \text{ is constant}
\][/tex]
Adding them together:
[tex]\[
S(9) = 243 + 81 + 52 = 376
\][/tex]
Thus, the equilibrium price is \$376.
2. Calculate the Area Under the Supply Curve from [tex]\( q = 0 \)[/tex] to [tex]\( q = 9 \)[/tex]:
To find the area under the supply curve, we need to integrate [tex]\( S(q) \)[/tex] from [tex]\( q = 0 \)[/tex] to [tex]\( q = 9 \)[/tex].
[tex]\[
\int_0^9 (q^{5/2} + 3q^{3/2} + 52) \, dq
\][/tex]
Compute the integral term-by-term:
[tex]\[
\int q^{5/2} \, dq = \frac{2}{7} q^{7/2}
\][/tex]
[tex]\[
\int 3q^{3/2} \, dq = \frac{6}{5} q^{5/2}
\][/tex]
[tex]\[
\int 52 \, dq = 52q
\][/tex]
Evaluate each at the bounds:
[tex]\[
\left[ \frac{2}{7} q^{7/2} \right]_0^9 = \frac{2}{7} (9^{7/2} - 0) = \frac{2}{7} (2187) = 624.8571
\][/tex]
[tex]\[
\left[ \frac{6}{5} q^{5/2} \right]_0^9 = \frac{6}{5} (9^{5/2} - 0) = \frac{6}{5} (243) = 291.6
\][/tex]
[tex]\[
\left[ 52q \right]_0^9 = 52 (9 - 0) = 468
\][/tex]
Adding these areas together gives:
[tex]\[
624.8571 + 291.6 + 468 = 1384.4571
\][/tex]
3. Calculate the Producer's Surplus:
The producer's surplus is the area under the supply curve minus the total revenue at the equilibrium price (equilibrium price times equilibrium quantity).
Total revenue at equilibrium is:
[tex]\[
\text{Equilibrium Price} \times \text{Equilibrium Quantity} = 376 \times 9 = 3384
\][/tex]
The producer's surplus is thus:
[tex]\[
1384.4571 - 3384 = -1999.5429
\][/tex]
Rounding to the nearest hundredth:
[tex]\[
-1999.54
\][/tex]
Therefore, the producer's surplus is:
[tex]\(\boxed{-1999.54}\)[/tex]
