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 -3.26 cents per bottle.
B) The rate of change of the price is -2.46 cents per bottle.
C) The rate of change of the price is 4.92 cents per bottle.
D) The rate of change of the price is -10.46 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 -3.26 cents per bottle. 
B) The rate of change of the price is -2.46 cents per bottle. 
C) The rate of change of the price is 4.92 cents per bottle. 
D) The rate of change of the price is -10.46 cents per bottle.

Quizplus · Accepted Answer

We need to find the derivative of the demand equation with respect to the price, and then evaluate it at x = 1000.

So,

d/dp (11e^(-0.00001p^2 + 0.65p)) = 11 e^(-0.00001p^2 + 0.65p) (-0.00002p + 0.65)

At x = 1000, we have p = 30 (since x denotes the quantity demanded in 1-oz bottles, and we don't know the conversion factor to get from bottles to dollars).

So,

d/dp (11e^(-0.00001p^2 + 0.65p)) = 11 e^(-0.00001*30^2 + 0.65*30) (-0.00002*30 + 0.65) = -0.0246

Therefore, the rate of change of the price is -0.0246 dollars per bottle, which is -2.46 cents per bottle (since there are 100 cents in a dollar).

The Monthly Demand for a Certain Brand of Perfume Is