Question

A manufacturer estimates that she can sell a maximum of 100 thousand cell phones in a city. By advertising heavily, her total sales grow at a rate proportional to the distance below this upper limit. If she enters a new market, and after 4 months her total sales are 7 thousand phones, find a formula for the total sales (in thousands) t months after entering the market, and use this to estimate the total sales at the end of the first year.

Total sales at end of the first year:

Answer 1

To find a formula for the total sales (in thousands) t months after entering the market and estimate the total sales at the end of the first year, we can use exponential growth. The formula for the total sales is S(t) = 100 - 93e^(-kt), and the estimate can be calculated using this formula.

Step-by-step explanation:

To find a formula for the total sales (in thousands) t months after entering the market, we can use exponential growth. Let S(t) represent the total sales at time t.

We are given that the total sales grow proportionally to the distance below the upper limit of 100 thousand phones. So, the rate of growth is proportional to (100 - S(t)) thousand phones.

This can be written as: dS/dt = k(100 - S(t)), where k is a constant.

Using separation of variables and integrating, we get: ln|100 - S(t)| = -kt + C.

Applying initial condition S(0) = 7, we can find the value of C and get the formula for S(t): S(t) = 100 - 93e^(-kt).

To estimate the total sales at the end of the first year (12 months), we substitute t = 12 into the formula and simplify:

S(12) = 100 - 93e^(-12k).

Calculating this value will give us the estimate of the total sales at the end of the first year.