Sure, let's solve this step-by-step.
We are given that the amount [tex]\( A(t) \)[/tex] of cesium-137 remaining after [tex]\( t \)[/tex] years can be calculated using the exponential decay formula:
[tex]\[
A(t) = 381 \left(\frac{1}{2}\right)^{\frac{t}{30}}
\][/tex]
where:
- [tex]\( 381 \)[/tex] grams is the initial amount of cesium-137.
- [tex]\( \frac{1}{2} \)[/tex] represents the fraction of the substance remaining after each half-life period of 30 years.
- [tex]\( t \)[/tex] is the elapsed time in years.
We need to find the remaining amount of cesium-137 after 20 years and after 100 years.
1. Amount after 20 years:
Substitute [tex]\( t = 20 \)[/tex] into the formula:
[tex]\[
A(20) = 381 \left(\frac{1}{2}\right)^{\frac{20}{30}}
\][/tex]
Calculate the exponent:
[tex]\[
\frac{20}{30} = \frac{2}{3}
\][/tex]
Plugging this into the equation:
[tex]\[
A(20) = 381 \left(\frac{1}{2}\right)^{\frac{2}{3}}
\][/tex]
Evaluate the expression:
[tex]\[
A(20) \approx 240 \text{ grams}
\][/tex]
Therefore, after 20 years, the remaining amount of cesium-137 is approximately [tex]\( 240 \)[/tex] grams.
2. Amount after 100 years:
Substitute [tex]\( t = 100 \)[/tex] into the formula:
[tex]\[
A(100) = 381 \left(\frac{1}{2}\right)^{\frac{100}{30}}
\][/tex]
Calculate the exponent:
[tex]\[
\frac{100}{30} \approx 3.33
\][/tex]
Plugging this into the equation:
[tex]\[
A(100) = 381 \left(\frac{1}{2}\right)^{3.33}
\][/tex]
Evaluate the expression:
[tex]\[
A(100) \approx 38 \text{ grams}
\][/tex]
Therefore, after 100 years, the remaining amount of cesium-137 is approximately [tex]\( 38 \)[/tex] grams.
To summarize:
- The amount of cesium-137 remaining after 20 years is [tex]\( \boxed{240} \)[/tex] grams.
- The amount of cesium-137 remaining after 100 years is [tex]\( \boxed{38} \)[/tex] grams.
