Given that carbon: `14("C"_(14) )` decays at a constant rate In such a way that it reduces to 50% in 5568 yr. Then, the age of an old wooden piece In which the carbon Is only 12.5% of the original Is equal to

Options

A. ₹ 16704
B. ₹ 16705
C. ₹ 16604
D. ₹ 16606

Answer

To solve the problem of determining the age of an old wooden piece in which the carbon-14 is only 12.5% of the original, we can follow these steps:

### Step 1: Understand the Decay Process
Carbon-14 decays at a constant rate, reducing to 50% of its original amount in 5568 years. This is known as its half-life.

**Hint:** Remember that 12.5% is equivalent to \( \frac{1}{8} \) of the original amount.

### Step 2: Determine the Relationship Between Decay and Time
Since the carbon-14 reduces to 50% in 5568 years, we can establish that:
- After 1 half-life (5568 years), \( n = \frac{n_0}{2} \)
- After 2 half-lives (2 × 5568 years), \( n = \frac{n_0}{4} \)
- After 3 half-lives (3 × 5568 years), \( n = \frac{n_0}{8} \)

**Hint:** Each half-life reduces the amount of carbon-14 by half.

### Step 3: Calculate the Total Time for 12.5% Remaining
Since 12.5% is \( \frac{1}{8} \) of the original amount, we need to find out how many half-lives it takes to reach this amount:
- \( n = \frac{n_0}{8} \) means we have gone through 3 half-lives.

**Hint:** Count the number of half-lives needed to reach \( \frac{1}{8} \).

### Step 4: Compute the Total Time
Now, we calculate the total time taken for 3 half-lives:
\[ \text{Total Time} = 3 \times \text{Half-life} = 3 \times 5568 \text{ years} \]

**Hint:** Multiply the number of half-lives by the duration of one half-life.

### Step 5: Perform the Calculation
\[ \text{Total Time} = 3 \times 5568 = 16704 \text{ years} \]

**Hint:** Ensure you perform the multiplication correctly to find the total time.

### Conclusion
The age of the old wooden piece is **16704 years**.