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. (a) 27190 years
B. (b) 1190 years
C. (c) 17190 years
D. (d) None of these

Explanation

To solve the problem step by step, we will follow the outlined reasoning in the video transcript and perform the necessary calculations.

### Step 1: Determine the initial decay rate for 200 grams of carbon-14.
The decay rate for a living organism is given as 15 decay/min/g. Therefore, for 200 grams:
\[ \text{Initial decay rate} (R_0) = 15 \, \text{decay/min/g} \times 200 \, \text{g} = 3000 \, \text{decay/min} \]

### Step 2: Set up the decay equation.
The decay rate of the bone is given as 375 decay/min. We can express the decay rate after \( n \) half-lives using the formula:
\[ R = R_0 \left(\frac{1}{2}\right)^n \]
Where:
- \( R \) is the current decay rate (375 decay/min),
- \( R_0 \) is the initial decay rate (3000 decay/min),
- \( n \) is the number of half-lives that have passed.

### Step 3: Substitute the values into the decay equation.
Substituting the known values into the equation:
\[ 375 = 3000 \left(\frac{1}{2}\right)^n \]

### Step 4: Solve for \( n \).
Rearranging the equation gives:
\[ \left(\frac{1}{2}\right)^n = \frac{375}{3000} \]
Calculating the right side:
\[ \frac{375}{3000} = \frac{1}{8} \]
Thus, we have:
\[ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 \]
This implies:
\[ n = 3 \]

### Step 5: Calculate the elapsed time.
The elapsed time since the death of the living organism can be calculated using the formula:
\[ \text{Elapsed time} = n \times t_{1/2} \]
Where \( t_{1/2} \) is the half-life of carbon-14, which is given as 5730 years. Therefore:
\[ \text{Elapsed time} = 3 \times 5730 \, \text{years} = 17190 \, \text{years} \]

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

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