Applied Calculus Question (Maximizing Revenue)?

Question

Here is the question (it deals with maximizing revenue):

According to Mr. Hass, the number of cell phone subscribers in North America was 150 million as of the year 2000 and he claims that the number of subscribers has been increasing at a rate of 42 million per year. He also claims the average annual revenue (to cell phone service providers) per subscriber was $330 in 2000 , but has been decreasing at a continuous rate of 5% per year since then. If this is all true, determine the year when cell phone revenue is expected to peak.

Answer 1

Let y be the number of years since 2000.

Let S(y) be the number (in millions) of subscribers y years after 2000. Then
S(y) = 150 + 42y

Let p(y) be the avg revenue (in dollars) per subscriber, y years after 2000. Then
p(y) = 330*0.95^y

Total revenue equals the number of subscribers, times avg revenue per subscriber. So the revenue R(y) (in millions of dollars), y years after 2000, is

R(y) = S(y)*p(y) = (150 + 42y)(330*0.95^y)

Maximize R.

I get y ≈ 15.9.

If we now assume that y must be an integer, we would need to look at R(15) and R(16); whichever is larger would correspond to the year in which revenue is expected to peak.