Question

A particular commodity has a price-demand equation given by p = √18,201-413x, where x is the amount in pounds of the commodity demanded when the price is p dollars

per pound.
(a) Find consumers' surplus if the equilibrium quantity is 40 pounds. (Round your answer to the nearest cent if necessary.)
(b) Find consumers' surplus if the equilibrium price is 17 dollars, (Round your answer to the nearest cent if necessary.)

Answer 1

The consumers' surplus is 0 dollars for both cases.

Step-by-step explanation:

To find the consumers' surplus, we need to calculate the area between the demand curve and the equilibrium quantity. In this case, the equilibrium quantity is 40 pounds. Substituting this value into the demand equation, we can find the equilibrium price:

p = √(18,201 - 413x) = √(18,201 - 413(40)) = √(18,201 - 16,520) = √681 = 26.09

The consumers' surplus is the area between the price (26.09) and the equilibrium price (26.09) multiplied by the quantity (40 pounds):

Consumers' surplus = (26.09 - 26.09) * 40 = 0

Therefore, the consumers' surplus is 0 dollars.

To find the consumers' surplus if the equilibrium price is 17 dollars, we need to find the equilibrium quantity. Substituting the price into the demand equation, we can solve for x:

17 = √(18,201 - 413x)

Squaring both sides, we get:

289 = 18,201 - 413x

413x = 17,912

x = 43.36

The consumers' surplus is the area between the price (17) and the equilibrium price (17) multiplied by the quantity (43.36 pounds):

Consumers' surplus = (17 - 17) * 43.36 = 0

Therefore, the consumers' surplus is 0 dollars.