A radioactive material decays according to the function $$ \mathrm{A}(\mathrm{t})=\mathrm{A}_{0} e^{-0.0244 t} $$, where $$ \mathrm{A}_{0} $$ is the initial amount present and A is the amount present at time $$ t $$ (in years). Assume that a scientist has a sample of 500 grams of the radioactive materal. (a) What is the decay rate of the radioactive material? (b) Graph the function using a graphing utility. (c) How much radioactive material is left after 30 years? (d) When will only 400 grams of the radioactive material be left? (e) What is the half-life of the radioactive material?

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Let's analyze the radioactive decay problem step by step.

Given the decay function:
$$
A(t) = A_0 e^{-0.0244 t}
$$
where:
- $$ A(t) $$ is the amount of radioactive material at time $$ t $$ (in years),
- $$ A_0 = 500 $$ grams is the initial amount.

---

### **(a) Decay Rate of the Radioactive Material**

**Decay Rate Definition:**
The decay rate refers to the constant rate at which a radioactive substance decreases over time. It is represented by the decay constant $$ k $$ in the exponential decay formula:
$$
A(t) = A_0 e^{-kt}
$$

**Given:**
$$
k = 0.0244 \ \text{year}^{-1}
$$

**Interpretation:**
- **Decay Constant ($$ k $$)**: 0.0244 per year.
- **Decay Rate as a Percentage:** To express the decay rate as a percentage:
  $$
  \text{Decay Rate} = k \times 100\% = 0.0244 \times 100\% = 2.44\% \ \text{per year}
  $$

**Answer:**
The decay rate of the radioactive material is **0.0244 per year** or **2.44% per year**.

---

### **(b) Graphing the Decay Function**

To graph the decay function $$ A(t) = 500 e^{-0.0244 t} $$, you can use various graphing utilities such as:
- **Graphing Calculators** (e.g., TI-84)
- **Online Tools** (e.g., Desmos, GeoGebra)
- **Spreadsheet Software** (e.g., Microsoft Excel, Google Sheets)

**Steps to Graph Using Desmos:**
1. **Access Desmos:** Go to [Desmos Graphing Calculator](https://www.desmos.com/calculator).
2. **Enter the Function:** In the input field, type `A(t) = 500 * e^(-0.0244 * t)`.
3. **Set the Domain:** Since time $$ t $$ is in years and the amount decreases over time, you might set $$ t $$ from 0 to, say, 50 years for a clear view.
4. **Adjust Axes:** Ensure the y-axis covers from 0 to slightly above 500 grams, and the x-axis covers the chosen time range.
5. **Plot the Graph:** Desmos will automatically plot the decay curve showing the exponential decrease of the radioactive material over time.

**Sample Graph Description:**
- **Y-axis (Amount in grams):** Starts at 500 and decreases towards 0.
- **X-axis (Time in years):** Shows the progression of time.
- **Curve:** A smooth, downward-sloping exponential curve representing the decay of the material.

---

### **(c) Amount of Radioactive Material Left After 30 Years**

**Given:**
$$
t = 30 \ \text{years}
$$

**Calculation:**
$$
A(30) = 500 \cdot e^{-0.0244 \times 30}
$$
$$
= 500 \cdot e^{-0.732}
$$
$$
\approx 500 \cdot 0.4805
$$
$$
\approx 240.25 \ \text{grams}
$$

**Answer:**
After **30 years**, approximately **240.25 grams** of the radioactive material remain.

---

### **(d) Time Until Only 400 Grams Remain**

**Given:**
$$
A(t) = 400 \ \text{grams}
$$
$$
A(t) = 500 \cdot e^{-0.0244 t}
$$

**Set Up the Equation:**
$$
400 = 500 \cdot e^{-0.0244 t}
$$
$$
\frac{400}{500} = e^{-0.0244 t}
$$
$$
0.8 = e^{-0.0244 t}
$$

**Solving for $$ t $$:**
$$
\ln(0.8) = -0.0244 t
$$
$$
-0.22314 = -0.0244 t
$$
$$
t = \frac{0.22314}{0.0244}
$$
$$
t \approx 9.146 \ \text{years}
$$

**Answer:**
It will take approximately **9.15 years** for only **400 grams** of the radioactive material to remain.

---

### **(e) Half-Life of the Radioactive Material**

**Half-Life Definition:**
The half-life is the time required for half of the radioactive substance to decay.

**Given:**
$$
A(t) = \frac{A_0}{2} = 500 \cdot e^{-0.0244 t}
$$
$$
250 = 500 \cdot e^{-0.0244 t}
$$
$$
\frac{250}{500} = e^{-0.0244 t}
$$
$$
0.5 = e^{-0.0244 t}
$$

**Solving for $$ t $$:**
$$
\ln(0.5) = -0.0244 t
$$
$$
-0.693147 = -0.0244 t
$$
$$
t = \frac{0.693147}{0.0244}
$$
$$
t \approx 28.37 \ \text{years}
$$

**Answer:**
The **half-life** of the radioactive material is approximately **28.37 years**.

---

**Summary of Answers:**
- **(a)** Decay rate is **0.0244 per year** or **2.44% per year**.
- **(b)** Graph the function $$ A(t) = 500 e^{-0.0244 t} $$ using a graphing utility like Desmos.
- **(c)** After **30 years**, approximately **240.25 grams** remain.
- **(d)** It takes about **9.15 years** for the amount to decrease to **400 grams**.
- **(e)** The half-life of the material is approximately **28.37 years**.
- **(a)** The decay rate is 0.0244 per year or 2.44% per year. - **(b)** Graph the function $$ A(t) = 500 e^{-0.0244 t} $$ using a graphing tool. - **(c)** After 30 years, about 240.25 grams remain. - **(d)** It takes approximately 9.15 years for the amount to decrease to 400 grams. - **(e)** The half-life is about 28.37 years.

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