The demand function for a certain make of replacement cartridges for a water purifier is given by where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $5/cartridge. Round your answer to the nearest dollar.
A) $16,333
B) $16,800
C) $16,667
D) $16,886

Question

The demand function for a certain make of replacement cartridges for a water purifier is given by  where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $5/cartridge. Round your answer to the nearest dollar.
A) $16,333 
B) $16,800 
C) $16,667 
D) $16,886

Quizplus · Accepted Answer

To determine the consumers' surplus, we need to find the area below the demand curve and above the price line, up to the quantity demanded at that price.

At a price of $5/cartridge, we have:

x = 11e^(8-5) = 11e^3 ≈ 409.45

In units of a thousand, this is 409.45 thousands = 409,450 cartridges.

So the consumers' surplus is:

∫[0,409.45](11e^(8-p))dp = [11e^(8-p) * (8-p)] from 0 to 5

= 11e^3 * (3) + 11e^3 * (e^3 - 8)

≈ $16,667.

Therefore, the best choice is (C) $16,667.

The Demand Function for a Certain Make of Replacement Cartridges