To calculate the consumer's surplus, follow these steps:

Step 1: Define Consumer's Surplus Formula

Consumer's surplus is the area between the demand curve and the equilibrium price line, from $q = 0$ to $q = q_{\text{eq}}$. Mathematically: $\text{Consumer's Surplus} = \int_0^{q_{\text{eq}}} D(q) \, dq - \text{Price} \cdot q_{\text{eq}}$ where:

• $q_{\text{eq}} = 55$,
• $D(q) = \frac{4000}{(2q + 5)^3}$.

Step 2: Find the Equilibrium Price

At $q_{\text{eq}} = 55$, the demand function gives: $P_{\text{eq}} = D(55) = \frac{4000}{(2(55) + 5)^3} = \frac{4000}{(110 + 5)^3} = \frac{4000}{115^3}.$ Compute $115^3$: $115^3 = 115 \times 115 \times 115 = 1,522,875.$ $P_{\text{eq}} = \frac{4000}{1,522,875} \approx 0.002626.$

Step 3: Compute the Definite Integral

The integral of $D(q) = \frac{4000}{(2q + 5)^3}$ is computed as follows: $\int \frac{4000}{(2q + 5)^3} \, dq = -\frac{4000}{2(2q + 5)^2} + C = -\frac{2000}{(2q + 5)^2} + C.$

Evaluate this integral from $q = 0$ to $q = 55$:

1. At $q = 55$: $\left. -\frac{2000}{(2q + 5)^2} \right|_{q = 55} = -\frac{2000}{(2(55) + 5)^2} = -\frac{2000}{115^2}.$ $115^2 = 115 \times 115 = 13,225, \quad \therefore -\frac{2000}{115^2} = -\frac{2000}{13,225} \approx -0.1512.$

2. At $q = 0$: $\left. -\frac{2000}{(2q + 5)^2} \right|_{q = 0} = -\frac{2000}{(2(0) + 5)^2} = -\frac{2000}{25} = -80.$

Now compute the definite integral: $\int_0^{55} D(q) \, dq = (-0.1512) - (-80) \approx 79.8488.$

Step 4: Compute Consumer's Surplus

Finally, subtract the rectangle area ($P_{\text{eq}} \cdot q_{\text{eq}}$) from the integral: $\text{Consumer's Surplus} = 79.8488 - (0.002626 \cdot 55) \approx 79.8488 - 0.14443 \approx 79.70.$

Final Answer:

The consumer's surplus is approximately $79.70.

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Would you like a detailed breakdown or clarification on any step?

5 Related Questions:

1. How is consumer surplus affected if equilibrium quantity changes?
2. Can you explain the economic significance of consumer surplus?
3. How does the shape of the demand curve influence surplus calculations?
4. What happens to the surplus if the price decreases below equilibrium?
5. How is consumer surplus used in welfare economics?

Tip:

Always round intermediate steps to a few extra decimal places to avoid rounding errors in final answers.