Half - life period of `""^(14)C` is 5770 years . If and old wooden toy has 0.25% of activity of `""^(14)C` Calculate the age of toy. Fresh wood has 2% activity of `""^(14)C`.

Answer

To calculate the age of the wooden toy based on the activity of Carbon-14, we can follow these steps:

### Step 1: Understand the Initial and Current Activity Levels
- Fresh wood has an activity of 2% of Carbon-14.
- The activity of the old wooden toy is 0.25% of Carbon-14.

### Step 2: Determine the Fraction of Carbon-14 Remaining
- To find out how much Carbon-14 remains in the toy compared to fresh wood, we can set up the following ratio:
\[ \text{Fraction remaining} = \frac{\text{Current activity}}{\text{Initial activity}} = \frac{0.25\%}{2\%} = \frac{0.25}{2} = 0.125 \]

### Step 3: Relate the Fraction Remaining to Half-Lives
- The fraction remaining can also be expressed in terms of half-lives using the formula:
\[ \text{Fraction remaining} = \left(\frac{1}{2}\right)^n \]
where \( n \) is the number of half-lives that have passed.

- Setting this equal to the fraction we calculated:
\[ 0.125 = \left(\frac{1}{2}\right)^n \]

### Step 4: Solve for \( n \)
- We know that \( 0.125 = \frac{1}{8} = \left(\frac{1}{2}\right)^3 \), so:
\[ n = 3 \]

### Step 5: Calculate the Age of the Toy
- Each half-life of Carbon-14 is 5770 years. Therefore, the total age of the toy can be calculated as:
\[ \text{Age of the toy} = n \times \text{Half-life} = 3 \times 5770 \text{ years} = 17310 \text{ years} \]

### Final Answer
- The age of the wooden toy is **17310 years**.

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