The age of a specimen, t, is related to the daughter/parent ratio D/P by the equation

Options

A. `t = (1)/(lamda)" ln" (D)/(P)`
B. `t = (1)/(lamda) ln (1 + (P)/(D))`
C. `t = (1)/(lamda) ln (1 + (D)/(P))`
D. `t = (1)/(lamda) ln (2 + (P)/(D))`

Explanation

To solve the problem of determining the age of a specimen (t) in relation to the daughter/parent ratio (D/P), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Variables**:
- Let \( A_0 \) be the initial amount of the parent isotope.
- Let \( A_t \) be the amount of the parent isotope remaining after time \( t \).
- Let \( D \) be the amount of the daughter isotope produced.
- The relationship between the daughter and parent isotopes can be expressed as:
\[ D = A_0 - A_t \]

2. **Establishing the Daughter/Parent Ratio**:
- The daughter/parent ratio \( \frac{D}{P} \) can be defined as:
\[ \frac{D}{P} = \frac{D}{A_t} \]
- Substituting \( D \) from the previous step:
\[ \frac{D}{P} = \frac{A_0 - A_t}{A_t} \]

3. **Rearranging the Ratio**:
- Rearranging the above equation gives:
\[ \frac{D}{P} = \frac{A_0}{A_t} - 1 \]
- Therefore, we can express \( \frac{A_0}{A_t} \) in terms of \( \frac{D}{P} \):
\[ \frac{A_0}{A_t} = \frac{D}{P} + 1 \]

4. **Using the Decay Constant**:
- The decay of the parent isotope can be described by the equation:
\[ A_t = A_0 e^{-\lambda t} \]
- Rearranging gives:
\[ \frac{A_0}{A_t} = e^{\lambda t} \]

5. **Equating the Two Expressions**:
- From the two expressions for \( \frac{A_0}{A_t} \):
\[ e^{\lambda t} = \frac{D}{P} + 1 \]

6. **Taking the Natural Logarithm**:
- Taking the natural logarithm of both sides:
\[ \lambda t = \ln\left(\frac{D}{P} + 1\right) \]

7. **Solving for Time (t)**:
- Finally, solving for \( t \):
\[ t = \frac{1}{\lambda} \ln\left(\frac{D}{P} + 1\right) \]

### Final Result:
The age of the specimen \( t \) is given by:
\[ t = \frac{1}{\lambda} \ln\left(\frac{D}{P} + 1\right) \]