MAT 210 optimization

Question

The number of cell phone subscribers in a certain country in the years 2000–2005 was projected to follow the equation N(t) = 39t + 74 million subscribers in year t (t = 0 represents January 2000). The average annual revenue per cell phone user was $350 in 2000. If we assume that due to competition the revenue per cell phone user decreases continuously at an annual rate of 20%, we can model the annual revenue as R(t) = 350(39t + 74)e^−0.2t million dollars.



(a) Determine when to the nearest 0.1 year the revenue was projected to peak. (Submit your answer in terms of t.

(b)Determine the revenue, to the nearest $1 million, at that time.

Answer 1

The point at which R(t) peaks is its maximum point. To find that, we take the derivative of R(T), set it to zero to find the critical point(s), and show that point is a maximum (as opposed to a minimum). Thus:

R(t)=350(39t+74)e^-0.2t

R’(t) = 350(-7.8xe^-0.2t+24.2e^-0.2t)

Solving R'(t)=0:

350(-7.8xe^-0.2t+24.2e^-0.2t)=0

T≈ 3.1

To prove this is a maximum, choose a point left of 3.1, 3.0, and another point right of 3.1, 3.2.

R’(3.0)=+153.67 >0 so R(t) is rising left of point 3.1.

R’(3.2)=-140.26<0 so R(t) is falling right of point 3.1. Thus, t=3.1 is a maximum.

Then:

R(3.1)≈$36696 million