The radioactive substance cesium-137 has a half-life of 30 years. The amount ( ) (in grams) of a sample of cesium-137 remaining after years is given by the
following exponental function.

Find the initial amount in the sample and the amount remaining after 80 years.
Round your answers to the nearest gram as necessary.

Ask by Byrd Perkins. in the United States

Mar 11,2025

Question

The radioactive substance cesium-137 has a half-life of 30 years. The amount ( ) (in grams) of a sample of cesium-137 remaining after years is given by the
following exponental function.

Find the initial amount in the sample and the amount remaining after 80 years.
Round your answers to the nearest gram as necessary.

Ask by Byrd Perkins. in the United States

Mar 11,2025

UpStudy AI · Accepted Answer

The decay function is given by

$$
A(t)=647\left(\frac{1}{2}\right)^{\frac{t}{30}}.
$$

**Step 1. Find the initial amount.**

When $$ t = 0 $$,

$$
A(0)=647\left(\frac{1}{2}\right)^{\frac{0}{30}} = 647\left(\frac{1}{2}\right)^{0} = 647 \cdot 1 = 647.
$$

Thus, the initial amount is $$ 647 $$ grams.

**Step 2. Find the amount after 80 years.**

Substitute $$ t = 80 $$ into the formula:

$$
A(80)=647\left(\frac{1}{2}\right)^{\frac{80}{30}}.
$$

Simplify the exponent:

$$
\frac{80}{30} = \frac{8}{3}.
$$

So,

$$
A(80)=647\left(\frac{1}{2}\right)^{\frac{8}{3}}.
$$

This can also be written as:

$$
A(80)=647 \cdot 2^{-\frac{8}{3}}.
$$

**Step 3. Evaluate $$ 2^{-\frac{8}{3}} $$.**

We can compute $$ 2^{-\frac{8}{3}} $$ approximately using logarithms or a calculator:

$$
2^{-\frac{8}{3}} \approx 0.1576.
$$

**Step 4. Compute $$ A(80) $$.**

Multiply the initial amount by the decay factor:

$$
A(80) \approx 647 \times 0.1576 \approx 102.0.
$$

Rounding to the nearest gram, after 80 years the sample has about $$ 102 $$ grams remaining.

**Final Answers:**
- Initial amount: $$ 647 $$ grams
- Amount after 80 years: $$ 102 $$ grams
The initial amount is 647 grams, and after 80 years, there are approximately 102 grams remaining.

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