Question

An investment loses 2.3​% of its value every week. find the​ half-life in this situation.

Answer 1

The half-life would be 29.79 weeks.

The amount of an investment is given by


`A=p(1+r)^t`

where A is the total amount, p is the amount invested, r is the interest rate as a decimal number, and t is the amount of time. In our case, we do not know the principal; r is -2.3% = -2.3/100 = -0.023; and we do not know t. We do know that we are trying to find when A, the amount of the investment, is equal to 1/2 of the principal; that means A = 0.5p:


`0.5p = p(1-0.023)^t`

We can divide both sides by p, which will cancel it on both sides:

`0.5 = (1-0.023)^t \\ \\0.5=0.977^t`

To cancel the exponent t, we will use logarithms:


`log_(0.977)0.5=t \\ \\t=29.789\approx 29.79`