Question

7. A radioactive substance is decaying such that 2% of its mass is lost every year. Originally there were 50kilograms of the substance present.(a) Write an equation for the amount, A, of thesubstance left after t-years.

Answer 1

If the decay ratio is 2% each year, there is left 98%, which is 0.98.

Then, if the initial mass is 50 kilograms, then we can express the following

`A=50(0.98)^t`

Then, we solve for t, when A = 25, which is half of the initial amount.

`\begin{gathered} 25=50\cdot(0.98)^t \\ (25)/(50)=(0.98)^t \\ (1)/(2)=(0.98)^t \end{gathered}`

Now, we use logarithms to find the value of t

`\begin{gathered} \log ((1)/(2))=\log (0.98)^t \\ \log ((1)/(2))=t\cdot\log (0.98) \\ t=(\log ((1)/(2)))/(\log (0.98)) \\ t\approx34.3 \end{gathered}`

Hence, the time that it takes for only half of the initial amount is 34.3 years.