Step 1: Define the demand function and market equilibrium.
The demand function is specified as: \[P = 110 - Q^2\]At market equilibrium, the price is \( P_0 = 29 \) and the quantity is \( Q_0 = 9 \).
Step 2: Determine the price at \( Q_0 = 9 \).
Substitute \( Q_0 = 9 \) into the demand function: \[P_0 = 110 - (9)^2 = 110 - 81 = 29\]This confirms the equilibrium price is \( P_0 = 29 \).
Step 3: Consumer Surplus Formula.
Consumer surplus is calculated as the area between the demand curve and the price level up to the equilibrium quantity. The formula is:\[\text{Consumer Surplus} = \frac{1}{2} \times \text{Base} \times \text{Height}\]The base is the equilibrium quantity \( Q_0 \). The height is the difference between the maximum price at \( Q = 0 \) and the equilibrium price \( P_0 \).
Step 4: Calculate the maximum price at \( Q = 0 \).
Substitute \( Q = 0 \) into the demand function: \[P = 110 - (0)^2 = 110\]The maximum price consumers will pay at \( Q = 0 \) is \( P = 110 \).
Step 5: Calculate the consumer surplus.
Apply the consumer surplus formula with the determined values: \[\text{Consumer Surplus} = \frac{1}{2} \times Q_0 \times (P_{\text{max}} - P_0)\]\[\text{Consumer Surplus} = \frac{1}{2} \times 9 \times (110 - 29)\]\[\text{Consumer Surplus} = \frac{1}{2} \times 9 \times 81\]\[\text{Consumer Surplus} = \frac{1}{2} \times 729 = 364.5\]
Final Answer: \[\boxed{364.5}\]