A bone containing `200 g` carbon-14 has `beta`-decay rate of `375` decay/min. Calculate the time that has elapsed since the death of the living one. Given the rate of decay for the living organism is equal to `15` decay per min per gram of carbon and half-life of carbon-14 is `5730 years`.

Options

A. 22920 yr
B. 11460 yr
C. 17190 yr
D. None of these

Explanation

To solve the problem, we need to calculate the time that has elapsed since the death of the living organism based on the decay of carbon-14. Let's break it down step by step.

### Step 1: Calculate the Initial Decay Rate (R0)
The decay rate for a living organism is given as 15 decay/min/g of carbon. The bone contains 200 g of carbon-14.

\[ R_0 = \text{decay rate per gram} \times \text{total grams of carbon} \]
\[ R_0 = 15 \, \text{decay/min/g} \times 200 \, \text{g} = 3000 \, \text{decay/min} \]

### Step 2: Identify the Current Decay Rate (R)
The current decay rate of the bone is given as:
\[ R = 375 \, \text{decay/min} \]

### Step 3: Use the Decay Formula
The relationship between the initial decay rate \( R_0 \), the current decay rate \( R \), and the number of half-lives \( n \) is given by the formula:
\[ R = R_0 \left(\frac{1}{2}\right)^n \]
Substituting the known values:
\[ 375 = 3000 \left(\frac{1}{2}\right)^n \]

### Step 4: Solve for n
Rearranging the equation to solve for \( n \):
\[ \left(\frac{1}{2}\right)^n = \frac{375}{3000} \]
\[ \left(\frac{1}{2}\right)^n = \frac{1}{8} \]
Since \( \frac{1}{8} = \left(\frac{1}{2}\right)^3 \), we find:
\[ n = 3 \]

### Step 5: Calculate the Total Time Elapsed
Each half-life of carbon-14 is given as 5730 years. Therefore, the total time elapsed since the death of the organism is:
\[ \text{Total time} = n \times \text{half-life} \]
\[ \text{Total time} = 3 \times 5730 \, \text{years} = 17190 \, \text{years} \]

### Final Answer
The time that has elapsed since the death of the living organism is **17190 years**.

---