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 find the consumers' surplus, we first need to determine the demand function's inverse, which gives us an expression for x in terms of p. We can do this by solving the demand function for p:
p = 11 - 12ln(x/1000)
Rearranging gives us:
ln(x/1000) = (11 - p)/12
Taking the exponential of both sides gives us:
x/1000 = e^((11-p)/12)
Multiplying both sides by 1000 gives us:
x = 1000e^((11-p)/12)
Now we can find the consumers' surplus by integrating the demand curve from 0 to the price at which it is set (in this case, $5 per cartridge):
CS = ∫[0,5] (11 - 12ln(x/1000)) dx
Substituting our expression for x in terms of p gives us:
CS = ∫[0,5] (11 - 12ln(e^((11-p)/12))) (1000e^((11-p)/12)) dp
Simplifying gives us:
CS = ∫[0,5] (11000e^(p/12) - 12000pe^(p/12)) dp
Evaluating the integral gives us:
CS = [11000/12 e^(p/12) - 12000/144 e^(p/12) - 12000/144 p e^(p/12)] [0,5]
CS = [916.67 e^(5/12) - 83.33 e^(5/12) - (20/3) e^(5/12)] - [916.67 - 83.33 - (20/3)]
CS ≈ $16,667 (rounded to the nearest dollar)

The Demand Function for a Certain Make of Replacement Cartridges