To find the amount of cesium-137 remaining after a certain number of years, we use the exponential decay function:
[tex]\[ A(t) = 381 \left( \frac{1}{2} \right)^{\frac{t}{30}} \][/tex]
### Step 1: Calculate the amount remaining after 40 years
We substitute [tex]\( t = 40 \)[/tex] into the function:
[tex]\[ A(40) = 381 \left( \frac{1}{2} \right)^{\frac{40}{30}} \][/tex]
Simplify the exponent:
[tex]\[ \frac{40}{30} = \frac{4}{3} \][/tex]
So we have:
[tex]\[ A(40) = 381 \left( \frac{1}{2} \right)^{\frac{4}{3}} \][/tex]
Next, we compute:
[tex]\[ \left( \frac{1}{2} \right)^{\frac{4}{3}} \approx 0.395 \][/tex]
Multiply this by the initial amount:
[tex]\[ A(40) \approx 381 \times 0.395 \approx 150.555 \][/tex]
Rounding to the nearest gram:
[tex]\[ A(40) \approx 151 \text{ grams} \][/tex]
### Step 2: Calculate the amount remaining after 80 years
We substitute [tex]\( t = 80 \)[/tex] into the function:
[tex]\[ A(80) = 381 \left( \frac{1}{2} \right)^{\frac{80}{30}} \][/tex]
Simplify the exponent:
[tex]\[ \frac{80}{30} \approx \frac{8}{3} \][/tex]
So we have:
[tex]\[ A(80) = 381 \left( \frac{1}{2} \right)^{\frac{8}{3}} \][/tex]
Next, we compute:
[tex]\[ \left( \frac{1}{2} \right)^{\frac{8}{3}} \approx 0.156 \][/tex]
Multiply this by the initial amount:
[tex]\[ A(80) \approx 381 \times 0.156 \approx 59.436 \][/tex]
Rounding to the nearest gram:
[tex]\[ A(80) \approx 60 \text{ grams} \][/tex]
### Final Results
- The amount after 40 years: [tex]\( \boxed{151} \)[/tex] grams
- The amount after 80 years: [tex]\( \boxed{60} \)[/tex] grams
