Rate of sales question?

Question

the projected rate of sales of zfones in the UK will initiallly be 50000 per month. it is projected that sales will double every two months as production and distribution improve. the product will be launched on 1 august 2007. assuming that the market grows as projected calculate...

1) the rate of sales at the end of december 2007

2) the date when sales reach a rate of 1500000 per month

Answer 1

let N = number of sales

dN/dt = kN

N = Noe^kt

No = 50,000

when t = 2 months, N = 100,000

100,000/50,000 = e^k2

[ln2]/2 = k ~ 0.3466

and

N = 50,000e^0.3466t where t = months

from this plug in the info to get the answers

1. December is month 4 so t = 4
and N = 50,000e^(0.3466*4)

2. [ln(1500000/50000)]/0.3466 = t month when 1500000 sold

Answer 2

We assume uniform growth, so that doubling every two months implies growth by 1- sqrt(2) = 41.4 % per month.
50000 * sqrt(2)^n, where n = months. After 12/07 =
50000 * sqrt(2)^5 = 200000*sqrt(2) = 282843.

then in (000) 50 * sqrt(2)^n = 1500, or sqrt(2)^n = 30, so n = 30/sqrt(2) = 21 (rounded). So sales reach 1.5 million/mo at May 2009.