Question

A certain element decays at a constant rate of 3% per year.If you start with 15 grams of the element, how long will it take before there are only three grams left?

Answer 1

Given the information on the problem, we can write the following function:

`v(t)=15\cdot(1-0.03)^t`

where v(t) denotes the weight of the element at time t.

Then, to find the time it will take to the element to weight only 3 grams, we have to solve v(t) = 3:

`\begin{gathered} v(t)=3 \\ \Rightarrow15(0.97)^t=3 \\ \Rightarrow(0.97)^t=(3)/(15)=(1)/(5) \\ \text{Applying natural logarithm on both sides:} \\ \ln (0.97^t)=\ln (0.2) \\ \Rightarrow t\cdot\ln (0.97)=\ln (0.2) \\ \Rightarrow t=(\ln (0.2))/(\ln (0.97))=52.8\approx53 \end{gathered}`

therefore, it will take approximately 53 years for the element to weight 3 grams