A radioactive isotope of sodium has a half-life of 15 hours. The table provides data from an experiment showing how the rate of decay of the isotope varies with time. The background count rate has not been subtracted from these data.

[tex]
\[
\begin{tabular}{|c|c|c|c|c|}
\hline
\text{Time (h)} & 0 & 10 & 20 & 30 \\
\hline
\text{Count rate (counts/s)} & 400 & 260 & 170 & 115 \\
\hline
\end{tabular}
\]
[/tex]

What is the background radiation count rate?

A. 12 counts/s
B. 15 counts/s
C. 20 counts/s
D. 30 counts/s

[tex]
\[
\begin{align}
& \frac{400-x}{4} = 115-x \\
& 400-x = 1460 \\
& 460-400 = 2 \\
& 60 = 3x \\
& x = \frac{60}{3}
\end{align}
\]
[/tex]

The graph shows the count rate from a radioactive source over a period of time.

You want to know the background radiation count if an isotope with a known half-life of 15 hours decays from 400 counts/s to 115 counts/s in 30 hours.

Background

While the isotope has decayed through 2 half lives, the background remains constant. For some background count x, we have ...

[tex]\left(\dfrac{1}{2}\right)^2(400 -x) = 115 -x \quad\Rightarrow\quad 3x=460-400 \quad\Rightarrow\quad x=20[/tex]

The background radiation count rate is 20 counts/s.