Question

Sales of a certain product are declining at a rate proportional to the amount of sales. If at the end of the first year the sales have declined by 22%, then how many years will have passed (since the beginning of the first year) when sales become only 31% of their original value? Express your answer as a decimal, correct to within 0.001 years.

Answer 1

The answer is "6.093 years".

Explanation:

The rate of decline in sales in
`22\%` per year.

The starting sales is 100 units:

Using compounding formula:

`\to 100 * (1-(22)/(100))^t=22\% \ of \ 100\\\\\to 100 * ((100-22)/(100))^t=(22)/(100) * \ 100\\\\\to ((78)/(100))^t=(22)/(100)\\\\\to 0.78^t=0.22\\\\\text{taking \log on both the sides}\\\\`

taking log on both sides

`\to \log_e \ 0.78^t= \log_e\ 0.22\\\\\to t \log_e \ 0.78= \log_e\ 0.22\\\\\to t = (\log_e 0.22)/(\log_e 0.78)\\\\`

`= (-0.6575)/(-0.1079)\\\\= (0.6575)/(0.1079)\\\\=6.093`