25) What percentage of a radioactive sample remains after 175.0 years if it has a half-life of 28.8 years? A. 0.230 B. 1.48 C 16.5 (D) 6.08

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To determine the percentage of a radioactive sample that remains after a certain period, you can use the **half-life formula**:

$$
\text{Remaining Percentage} = \left( \frac{1}{2} \right)^{\frac{t}{\text{half-life}}} \times 100\%
$$

**Given:**
- **Time elapsed (t):** 175.0 years
- **Half-life:** 28.8 years

**Step-by-Step Calculation:**

1. **Calculate the number of half-lives that have passed:**

$$
\frac{t}{\text{half-life}} = \frac{175.0\ \text{years}}{28.8\ \text{years}} \approx 6.076
$$

2. **Apply the half-life formula:**

$$
\text{Remaining Percentage} = \left( \frac{1}{2} \right)^{6.076} \times 100\%
$$

3. **Calculate $$ \left( \frac{1}{2} \right)^{6.076} $$:**

$$
\left( \frac{1}{2} \right)^6 = \frac{1}{64} \approx 0.015625 \quad (1.5625\%)
$$

Since 6.076 is slightly more than 6, the remaining percentage will be slightly less than 1.5625%. A more precise calculation gives approximately **1.48%**.

**Answer:**

**B. 1.48**
After 175.0 years, approximately 1.48% of the radioactive sample remains.

25) What percentage of a radioactive sample remains after 175.0 years if it has a half-life of 28.8 years? A. 0.230 B. 1.48 C 16.5 (D) 6.08

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