Sure, let's go through the problem step by step to understand how we can find the initial amount of the cesium-137 sample and the amount remaining after 80 years.
### Step 1: Understand the Exponential Decay Formula
The formula provided to model the decay of cesium-137 is:
[tex]\[
A(t) = 934 \left( \frac{1}{2} \right)^{\frac{t}{30}}
\][/tex]
where:
- [tex]\( A(t) \)[/tex] represents the amount of cesium-137 remaining after [tex]\( t \)[/tex] years.
- 934 is the initial amount of cesium-137.
- [tex]\( \frac{1}{2} \)[/tex] represents the decay factor because cesium-137 decays to half its amount every 30 years (its half-life).
### Step 2: Determine the Initial Amount in the Sample
To find the initial amount, we inspect the equation at [tex]\( t = 0 \)[/tex]:
[tex]\[
A(0) = 934 \left( \frac{1}{2} \right)^{\frac{0}{30}}
\][/tex]
Since any number raised to the power of 0 is 1:
[tex]\[
A(0) = 934 \left( \frac{1}{2} \right)^0 = 934 \times 1 = 934
\][/tex]
So, the initial amount is 934 grams.
### Step 3: Calculate the Amount Remaining After 80 Years
Next, we need to find [tex]\( A(80) \)[/tex], the amount of cesium-137 remaining after 80 years. Using the formula:
[tex]\[
A(t) = 934 \left( \frac{1}{2} \right)^{\frac{t}{30}}
\][/tex]
we plug in [tex]\( t = 80 \)[/tex]:
[tex]\[
A(80) = 934 \left( \frac{1}{2} \right)^{\frac{80}{30}}
\][/tex]
### Step 4: Simplify the Exponential Term
Calculate the exponent:
[tex]\[
\frac{80}{30} \approx 2.67
\][/tex]
Using the value of [tex]\(2.67\)[/tex] as the exponent for the decay factor [tex]\( \left( \frac{1}{2} \right) \)[/tex]:
[tex]\[
A(80) = 934 \left( \frac{1}{2} \right)^{2.67}
\][/tex]
### Step 5: Calculate the Amount
Evaluate [tex]\( \left( \frac{1}{2} \right)^{2.67} \)[/tex] and multiply by 934:
[tex]\[
A(80) \approx 934 \times 0.158
\][/tex]
This computation yields:
[tex]\[
A(80) \approx 147.43
\][/tex]
### Step 6: Round to the Nearest Gram
Round the amount to the nearest gram:
[tex]\[
A(80) \approx 147 \text{ grams}
\][/tex]
### Conclusion
Thus, after solving, we have the following results:
- Initial Amount: 934 grams
- Amount After 80 Years: 147 grams
So, the answers are:
Initial amount: 934 grams
Amount after 80 years: 147 grams
