Question

Since 2008, the net sales S of a coffee company have grown exponentially at the rate of 46.3% per year. This growth can be modeled by dSdt=0.463S, and the function S satisfying this equation is S=500000e0.463t. How many years will it take for the net sales to double? (Round to three decimal places, and don't include units)

Answer 1

1.497 years

Explanation:

`S=500000e^(0.463t)`

In 2008, t = 0

`S=500000e^(0.463* 0) = 500000`

When sales are doubled, S = 1000000

`1000000=500000e^(0.463t)`

`e^(0.463t) = 2`

`0.463t = \ln 2 = 0.693`

`t = (0.693)/(0.463) = 1.497`

Therefore, sales will double in 1.497 years

Answer 2

1.497

Explanation:

If net sales are given by the function:

`S=500,000*e^(0.463t)`

The initial net sales, at t = 0 are:

`S=500,000*e^(0.463*0)\\S=500,000`

In order for sales to double, then S(t) must be equal to 1,000,000. The time 't', in years, required for sales to double is:

`1,000,000=500,000*e^(0.463t)\\2=e^(0.463t)\\ln(2) = 0.463t\\t=1.497\ years`

It will take 1.497 years for net sales to double.