Question

In this problem, p is in dollars and x is the number of units. The demand function for a certain product is p = 100 − x2 and the supply function is p = x2 4x 84. Find the equilibrium point. (x, p) =(2,96) Find the consumer's surplus there. (Round your answer to the nearest cent.) $=(?)

Answer 1

a) The equilibrium point is (x, p) = (4, 84).

b) The consumer's surplus is 32.

Step-by-step explanation:

To find the equilibrium point, we need to set the demand function equal to the supply function and solve for x.

So, we have: 100 - x^2 = x^2 - 4x + 84

Rearranging the equation, we get: 2x^2 - 4x - 16 = 0

Using the quadratic formula, we find two possible values for x: -2 and 4. Since the number of units cannot be negative, we discard -2 and consider x = 4 as the equilibrium point.

Therefore, the equilibrium point is (x, p) = (4, 84).

The consumer's surplus is the area between the demand curve and the equilibrium price line.

To find it, we calculate the area of the triangle formed by the equilibrium point (4, 84) and the x-axis.

The base of the triangle is 4 units, and the height is the difference between the equilibrium price and the highest price on the demand curve, which is 100 - 84 = 16.

Therefore, the consumer's surplus is (1/2) * base * height

= (1/2) * 4 * 16

= 32.

Therefore, the consumer's surplus is $32.