Question

A radioactive isotope that was originally 275 grams has decayed to 150 grams. The equation shown can be used to calculate the number of years, t, it took for the decay to occur.



150 equals 275 e Superscript negative 0.1386 t Baseline

Question
What is the approximate number of years it took for the isotope to decay from 275 grams to 150 grams?

Answer options with 5 options
A.
0.1 years

B.
0.4 years

C.
1.9 years

D.
3.9 years

E.
4.4 years

Answer 1

To solve for the time \( t \) in the equation

\[
150 = 275 e^{-0.1386t}
\]

we can rearrange the equation:

1. Divide both sides by 275:

\[
\frac{150}{275} = e^{-0.1386t}
\]

2. Simplify the left side:

\[
0.545454545 = e^{-0.1386t}
\]

3. Take the natural logarithm of both sides:

\[
\ln(0.545454545) = -0.1386t
\]

4. Calculate the natural logarithm:

\[
\ln(0.545454545) \approx -0.606
\]

5. Now, solve for \( t \):

\[
-0.606 = -0.1386t
\]

\[
t = \frac{-0.606}{-0.1386} \approx 4.37 \text{ years}
\]

Thus, the approximate number of years it took for the isotope to decay from 275 grams to 150 grams is about **4.4 years**.

So the correct answer is:

**E. 4.4 years**