Question

It is estimated that the demand for a manufacturer's product is increasing exponentially at an instantaneous rate of 5% per year. If the current demand is increasing by 4000 units per year and if the price remains fixed at $850 per unit, how much revenue will the manufacturer receive from the sale of the product over the next 5 years?

Any help would be great, the book doesn't give any examples of this type of problem.

Answer 1

dn/dt = .05 n
dn/n = . 05 dt
ln n = .05 t + k

n = e^(.05 t+k) = e^k * e^.05 t = C e^.05 t

dn/dt = C(.05) e^.05t
when t = 0 dn/dt = 4000
so
4000 = .05 C (1)
C = 80,000
so
n = 80,000 e^.05 t
at t = 5, n = 80,000 e^.25
or
n = 102,722 units
102722 * 850 = 87,313,728.33