The monthly demand for a certain brand of perfume is given by the demand equation where p denotes the retail unit price (in dollars) and x denotes the quantity (in 1-oz bottles) demanded. Find the rate of change of the price to the nearest hundredth of a cent per bottle when x = 1,000.
A) The rate of change of the price is - 3.62 cents per bottle.
B) The rate of change of the price is 5.24 cents per bottle.
C) The rate of change of the price is - 2.62 cents per bottle.
D) The rate of change of the price is - 2.22 cents per bottle.

Question

The monthly demand for a certain brand of perfume is given by the demand equation  where p denotes the retail unit price (in dollars) and x denotes the quantity (in 1-oz bottles) demanded. Find the rate of change of the price to the nearest hundredth of a cent per bottle when x = 1,000.
A) The rate of change of the price is - 3.62 cents per bottle. 
B) The rate of change of the price is 5.24 cents per bottle. 
C) The rate of change of the price is - 2.62 cents per bottle. 
D) The rate of change of the price is - 2.22 cents per bottle.

Quizplus · Accepted Answer

To find the rate of change of the price, we need to differentiate the demand equation with respect to p:

dD/dp = -11(1/x)

When x = 1000, we have:

dD/dp = -11(1/1000) = -0.011

This tells us that for every 1 cent increase in price, the quantity demanded will decrease by 0.011 units. To find the rate of change in price per bottle, we can divide this by the quantity demanded at x = 1000:

-0.011/1000 = -0.000011 = -0.0011 cents per bottle

Rounding to the nearest hundredth of a cent (as stated in the problem), we get:

-0.0026 cents per bottle

Therefore, the correct choice is C.

The Monthly Demand for a Certain Brand of Perfume Is