Question

The number of employees at a firm is modelled by the function `f(x) = 500/(1+19e^(-0.6x))` where `x` is the number of years. Which of the following statements is true?

• The initial number of employees was `20`.

• The first year that the number of employees exceeded `100` was the third year.

• The rate of increase will continue to increase indefinitely.

• The number of employees will never exceed `500`.

Answer 1

false, true, false, true...

Explanation:

Given:

`f(x) = 500/(1+19e^(-0.6x))`

We find:

`f(0) = 500/(1+19e^0) = 500/20 = 25`

So the first statement is false: The initial number of employees was actually `25`, not `20`

`f(2) = 500/(1+19e^-1.2) ~~ 500/(1+5.722) ~~ 74.4`

`f(3) = 500/(1+19e^-1.8) ~~ 500/(1+3.14) ~~ 121`

So the second statement is true: The first year that the number of employees exceeded `100` was the third year.

Note that as `x->oo`, `19e^(-0.6x) -> 0` and `f(x)->500` monotonically.

So the rate of increase of `f(x)` cannot increase indefinitely. On the contrary it must slow down, so that the number never exceeds `500`.

So the third statement is false and the fourth statement is true.