To solve this problem, we need to use the exponential decay formula given by:
[tex]\[ A(t) = 381 \left(\frac{1}{2}\right)^{\frac{t}{30}} \][/tex]
where:
- [tex]\( A(t) \)[/tex] represents the amount of cesium-137 remaining after [tex]\( t \)[/tex] years.
- The initial amount (when [tex]\( t = 0 \)[/tex] years) is 381 grams.
- The half-life of cesium-137 is 30 years.
- [tex]\( t \)[/tex] is the time in years.
Step 1: Determine the initial amount in the sample
- The initial amount of cesium-137 in the sample is given directly in the formula.
- When [tex]\( t = 0 \)[/tex], the formula simplifies to:
[tex]\[ A(0) = 381 \left(\frac{1}{2}\right)^{\frac{0}{30}} \][/tex]
- Any number to the power of 0 is 1, so:
[tex]\[ A(0) = 381 \times 1 \][/tex]
[tex]\[ A(0) = 381 \][/tex]
Therefore, the initial amount of cesium-137 is 381 grams.
Step 2: Calculate the remaining amount after 100 years
- To find the amount remaining after 100 years, we substitute [tex]\( t = 100 \)[/tex] into the formula:
[tex]\[ A(100) = 381 \left(\frac{1}{2}\right)^{\frac{100}{30}} \][/tex]
- Simplifying the exponent:
[tex]\[ \frac{100}{30} \approx 3.3333 \][/tex]
- So, the equation becomes:
[tex]\[ A(100) = 381 \left(\frac{1}{2}\right)^{3.3333} \][/tex]
- Calculating [tex]\(\left(\frac{1}{2}\right)^{3.3333}\)[/tex], we get approximately 0.09954.
- Therefore:
[tex]\[ A(100) = 381 \times 0.09954 \][/tex]
[tex]\[ A(100) \approx 37.93734 \][/tex]
- Rounding to the nearest gram, we get:
[tex]\[ A(100) \approx 38 \][/tex]
Therefore, the remaining amount of cesium-137 after 100 years is 38 grams.
Summary:
- Initial amount: 381 grams
- Amount after 100 years: 38 grams
