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?

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

Answer 2

To find the revenue over the next 5 years, we need to determine the total number of units sold each year and then multiply it by the price per unit.

First, let's determine the number of units sold each year. We're given that the demand is increasing exponentially at a rate of 5% per year. This means that every year, the demand will increase by 5% compared to the previous year.

Let's denote the current demand as D0 and let D1, D2, D3, D4, and D5 represent the demands for years 1, 2, 3, 4, and 5 respectively.

We know that the initial demand D0 is increasing by 4000 units per year. So D1 = D0 + 4000.

Using the exponential growth formula, we can calculate the demands for the following years:

D2 = D1 + 0.05 * D1
D3 = D2 + 0.05 * D2
D4 = D3 + 0.05 * D3
D5 = D4 + 0.05 * D4

Now that we have the number of units sold for each year, we can calculate the revenue by multiplying the number of units sold by the price per unit ($850 in this case) for each year:

Year 1 Revenue = D1 * 850
Year 2 Revenue = D2 * 850
Year 3 Revenue = D3 * 850
Year 4 Revenue = D4 * 850
Year 5 Revenue = D5 * 850

Finally, we can sum up the revenues for each year to get the total revenue over the next 5 years:

Total Revenue = Year 1 Revenue + Year 2 Revenue + Year 3 Revenue + Year 4 Revenue + Year 5 Revenue

By substituting the values we calculated for each year, we can find the total revenue.