Question

Can you please help with this question about exponential equations?

Answer 1

Since
`t` represents the number of years passed since 2005, in 2005 we have
`t=0`, and in 2015 we have
`t=10`. Let
`P_J(y),P_D(y)` represent the profits of Jones and Davis, respectively, at year y.

We have

`P_J(2005) = 10(0.99)^0 = 10,\quad P_J(2015) = 10(0.99)^(10)\approx 9`

`P_D(2005) = 8(1.01)^0 = 8,\quad P_D(2015) = 8(1.01)^(10)\approx 9`

Choosing the best company somehow depends on how much time you can wait: Jones start with a higher value (10 vs 8), but it has a descending trend, because the multiplicative factor
`0.99^t` goes to zero as t grows.

On the other hand, Davis starts with a smaller value, but
`1.01^t` tends to infinity as t grows.

So, if you need an immediate result, the most valuable company is Jones, otherwise you're certain can Davis will eventually become bigger.

More precisely, we have

`8\cdot 1.01^t = 10\cdot 0.99^t \iff 0.8 = \left((0.99)/(1.01)\right)^t \iff t\approx 11`

So, after 11 years, Davis will overcome Jones.