Question

The monthly demand for a certain brand of perfume is given by the demand equation p = 100e−0.0002x + 125 where p denotes the retail unit price (in dollars) and x denotes the quantity (in 1-oz bottles) demanded.
(a) Find the rate of change of the price per bottle when x = 2000 and when x = 3000. (Round your answers to four decimal places.)

x = 2000( ) dollars/bottle

x = 3000( ) dollars/bottle

(b) What is the price per bottle when x = 2000? When x = 3000? (Round your answers to the nearest cent.)

x = 2000 $ ( )

x = 3000 $ ( )

Answer 1

Answer:

a)

Rate of change of the price per bottle when x = 2000

p'(2000)= -0.0134

Rate of change of the price per bottle when x = 3000

p'(3000)= -0.011

b)

Price per bottle when x = 2000

$192

Price per bottle when x = 3000

$180

Step-by-step explanation:

The equation of the demand x in terms of the price p is

$\bf p(x)=100e^{-0.0002x}+125$

(a) Find the rate of change of the price per bottle when x = 2000 and when x = 3000. (Round your answers to four decimal places.)

These are p'(x) at x=2000 and x=3000

$\bf p'(x)=100(-0.0002)e^{-0.0002x}=-0.02e^{-0.0002x}$

so

$\bf p'(2000)=-0.02e^{-0.0002*2000}=-0.02e^{-0.4}=-0.0134$

$\bf p'(3000)=-0.02e^{-0.0002*3000}=-0.02e^{-0.6}=-0.011$

(b) What is the price per bottle when x = 2000? When x = 3000? (Round your answers to the nearest cent.)

These are p(2000) and p(3000)

$\bf p(2000)=100e^{-0.0002*2000}+125=100e^{-0.4}+125\approx 192$

$\bf p(3000)=100e^{-0.0002*3000}+125=100e^{-0.6}+125\approx 180$