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
[tex] \bf p(x)=100e^{-0.0002x}+125[/tex]
(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
[tex] \bf p'(x)=100(-0.0002)e^{-0.0002x}=-0.02e^{-0.0002x}[/tex]
so
[tex] \bf p'(2000)=-0.02e^{-0.0002*2000}=-0.02e^{-0.4}=-0.0134[/tex]
[tex] \bf p'(3000)=-0.02e^{-0.0002*3000}=-0.02e^{-0.6}=-0.011[/tex]
(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)
[tex] \bf p(2000)=100e^{-0.0002*2000}+125=100e^{-0.4}+125\approx 192[/tex]
[tex] \bf p(3000)=100e^{-0.0002*3000}+125=100e^{-0.6}+125\approx 180[/tex]
