Questions: HW-6.2: Consumer's and Producer's Surplus Score: 2 / 8 Answered: 2 / 8 Question 3 The demand for a particular item is given by the demand function D(x) = 1000 - 3x^2 Find the consumer's surplus if the equilibrium point (xe, pe) = (10,700).

To find the consumer's surplus, we need to follow these steps:

1. Understand the demand function and equilibrium point: The demand function is given by \( D(x) = 1000 - 3x^2 \). The equilibrium point is \( (x_e, p_e) = (10, 700) \).

2. Determine the equilibrium price: At the equilibrium quantity \( x_e = 10 \), the price \( p_e \) is given as 700.

3. Set up the integral for consumer's surplus: Consumer's surplus is the area between the demand curve and the price level up to the equilibrium quantity. Mathematically, it is given by: \[ \text{Consumer's Surplus} = \int_{0}^{x_e} D(x) \, dx - p_e \cdot x_e \] Here, \( x_e = 10 \) and \( p_e = 700 \).

4. Calculate the integral of the demand function from 0 to 10: \[ \int_{0}^{10} (1000 - 3x^2) \, dx \] We can split this into two integrals: \[ \int_{0}^{10} 1000 \, dx - \int_{0}^{10} 3x^2 \, dx \]

   Evaluate each integral separately: \[ \int_{0}^{10} 1000 \, dx = 1000x \Big|_{0}^{10} = 1000 \cdot 10 - 1000 \cdot 0 = 10000 \] \[ \int_{0}^{10} 3x^2 \, dx = 3 \int_{0}^{10} x^2 \, dx = 3 \left( \frac{x^3}{3} \Big|_{0}^{10} \right) = x^3 \Big|_{0}^{10} = 10^3 - 0^3 = 1000 \]

   Therefore: \[ \int_{0}^{10} (1000 - 3x^2) \, dx = 10000 - 1000 = 9000 \]

5. Calculate the consumer's surplus: \[ \text{Consumer's Surplus} = 9000 - (700 \cdot 10) = 9000 - 7000 = 2000 \]

So, the consumer's surplus is \$2000.

Summary: The consumer's surplus at the equilibrium point \((10, 700)\) is \$2000.