Sure, let's go through the each part one by one:
### Part (a): Determine the Equilibrium Price and Quantity
To find the price at which the market for the car stereos is in equilibrium, we need to set the demand function equal to the supply function and solve for the quantity, [tex]\(q\)[/tex].
The given demand function is [tex]\(D(q) = 228.8 - 16q\)[/tex].
The given supply function is [tex]\(S(q) = 29q + 57.8\)[/tex].
Set the demand equal to the supply:
[tex]\[
228.8 - 16q = 29q + 57.8
\][/tex]
Combine like terms to solve for [tex]\(q\)[/tex]:
[tex]\[
228.8 - 57.8 = 29q + 16q
\][/tex]
[tex]\[
171 = 45q
\][/tex]
[tex]\[
q = \frac{171}{45} \approx 3.8
\][/tex]
Now, substitute [tex]\(q = 3.8\)[/tex] back into either the demand or supply function to find the equilibrium price. Let's use the demand function:
[tex]\[
D(3.8) = 228.8 - 16(3.8) = 228.8 - 60.8 = 168
\][/tex]
Equilibrium Quantity ([tex]\(q\)[/tex]) is 3.8 thousand units.
Equilibrium Price ([tex]\(P\)[/tex]) is \$168.
### Part (b): Compute the Total Surplus
Total surplus is the sum of the consumer surplus and producer surplus. To find these, we use the equilibrium price and quantity.
#### Consumer Surplus:
Consumer surplus is the area between the demand curve and the price level up to the equilibrium quantity.
[tex]\[
\text{Consumer Surplus} = \int_{0}^{3.8} (D(q) - 168) \, dq
\][/tex]
Where [tex]\(D(q) = 228.8 - 16q\)[/tex]:
[tex]\[
\int_{0}^{3.8} (228.8 - 16q - 168) \, dq
\][/tex]
[tex]\[
\int_{0}^{3.8} (60.8 - 16q) \, dq
\][/tex]
The consumer surplus calculates to approximately 115.52 thousand dollars.
#### Producer Surplus:
Producer surplus is the area between the supply curve and the price level up to the equilibrium quantity.
[tex]\[
\text{Producer Surplus} = \int_{0}^{3.8} (168 - S(q)) \, dq
\][/tex]
Where [tex]\(S(q) = 29q + 57.8\)[/tex]:
[tex]\[
\int_{0}^{3.8} (168 - 29q - 57.8) \, dq
\][/tex]
[tex]\[
\int_{0}^{3.8} (110.2 - 29q) \, dq
\][/tex]
The producer surplus calculates to approximately 209.38 thousand dollars.
#### Total Surplus:
The total surplus is the sum of consumer surplus and producer surplus:
[tex]\[
\text{Total Surplus} = 115.52 + 209.38 = 324.9
\][/tex]
Total surplus is 324.9 thousand dollars.
### Part (c): Maximum Total Surplus
The maximum total surplus given is 324900 thousand dollars.
This concludes the solution for the given problem: we have computed the equilibrium price and quantity, the consumer surplus, producer surplus, and the total surplus along with the maximum total surplus.
