It is known that the quantity demanded of a certain make of portable hair dryer is x hundred units/week and the corresponding wholesale unit price is

Question

p=√169-7x (169-7x is under the whole square root symbol)

dollars. Determine the consumers' surplus if the market price is set at $8/unit. (Round your answer to two decimal places.)



Answer 1

The answer is 40.48.

To solve this question, we figure out first the defined area of the consumer surplus. Consumer surplus is the integrated area under the price curve above the market price and through the clearing price/unit. In this case, the price is $8 -> 8 = sqrt(169-7x) -> 64 = 169-7x -> -105 = -7x -> 15 = x. So, the market clearing condition is p = 8, x = 15.

We then need to integrate the area under the curve through x = 0 to 15, and above p=8. Integral of sqrt(169-7x) from 0 to 15 is 160.48, and we subtract the area of the market clearing condition (8 x 15 = 120). So we get 40.48.

You can also change the integral to solve in one step - Integral of sqrt(169-7x)-8 from 0 to 15 is 40.48.