Ountrinic Epronamial und Lepricimic forctions
Evelhating an exponential function that models a feul worid sitimition

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

Find the initial amount in the sample and the amount remaining after 100 years.
Round your answers to the nearest gram as necessary.

Initial amount:
grams

Amount after 100 years: grams

Ask by Valdez Ayala. in the United States

Jan 23,2025

Question

Ountrinic Epronamial und Lepricimic forctions
Evelhating an exponential function that models a feul worid sitimition

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

Find the initial amount in the sample and the amount remaining after 100 years.
Round your answers to the nearest gram as necessary.

Initial amount:
grams

Amount after 100 years: grams

Ask by Valdez Ayala. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To solve the problem of determining the initial amount of cesium-137 in the sample and the amount remaining after 100 years, we'll use the given exponential decay function:

$$
A(t) = 621 \left(\frac{1}{2}\right)^{\frac{t}{30}}
$$

where:
- $$ A(t) $$ is the amount of cesium-137 remaining after $$ t $$ years.
- $$ t $$ is the time in years.
- 621 grams is the initial amount of cesium-137.

### 1. **Initial Amount in the Sample**

The initial amount is the amount of cesium-137 present at time $$ t = 0 $$.

$$
A(0) = 621 \left(\frac{1}{2}\right)^{\frac{0}{30}} = 621 \times 1 = 621 \text{ grams}
$$

**Initial amount:** **621 grams**

### 2. **Amount Remaining After 100 Years**

To find the amount remaining after 100 years, substitute $$ t = 100 $$ into the function:

$$
A(100) = 621 \left(\frac{1}{2}\right)^{\frac{100}{30}} = 621 \left(\frac{1}{2}\right)^{3.\overline{3}}
$$

First, simplify the exponent:

$$
\frac{100}{30} = \frac{10}{3} \approx 3.3333
$$

Now, calculate $$ \left(\frac{1}{2}\right)^{3.3333} $$:

$$
\left(\frac{1}{2}\right)^{3.3333} = 2^{-3.3333} \approx 0.0992
$$

Multiply this by the initial amount:

$$
A(100) \approx 621 \times 0.0992 \approx 61.6 \text{ grams}
$$

Rounding to the nearest gram:

**Amount after 100 years:** **62 grams**

### **Summary**

- **Initial amount:** 621 grams
- **Amount after 100 years:** 62 grams
**Initial amount:** 621 grams **Amount after 100 years:** 62 grams

Upstudy AI Solution

Answer

Solution

1. Initial Amount in the Sample

2. Amount Remaining After 100 Years

Summary

The Deep Dive

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