The fraction of atoms of radioactive element that decays in 6 days is `(7)/(8)`. The fraction that decays in 10 days will be

Options

A. `(77)/(80)`
B. `(71)/(80)`
C. `(31)/(32)`
D. `(15)/(16)`

Answer

To solve the problem, we need to find the fraction of radioactive atoms that decays in 10 days, given that the fraction that decays in 6 days is \( \frac{7}{8} \).

### Step-by-step Solution:

1. **Determine the fraction of atoms left after 6 days:**
\[ \text{Fraction left} = 1 - \text{Fraction decayed} = 1 - \frac{7}{8} = \frac{1}{8} \]

2. **Use the decay formula to find the decay constant (\( \lambda \)):**
The decay of a radioactive substance can be described by the equation:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N(t) \) is the number of atoms remaining after time \( t \), \( N_0 \) is the initial number of atoms, and \( \lambda \) is the decay constant.

From the fraction left after 6 days:
\[ \frac{N(t)}{N_0} = e^{-\lambda t} = \frac{1}{8} \]
Setting \( t = 6 \) days, we have:
\[ e^{-\lambda \cdot 6} = \frac{1}{8} \]

3. **Take the natural logarithm of both sides:**
\[ -\lambda \cdot 6 = \ln\left(\frac{1}{8}\right) \]
Simplifying gives:
\[ \lambda = -\frac{\ln\left(\frac{1}{8}\right)}{6} = \frac{\ln(8)}{6} \]

4. **Calculate \( \ln(8) \):**
Since \( 8 = 2^3 \):
\[ \ln(8) = 3 \ln(2) \]
Using \( \ln(2) \approx 0.693 \):
\[ \ln(8) \approx 3 \times 0.693 \approx 2.079 \]
Therefore,
\[ \lambda \approx \frac{2.079}{6} \approx 0.3465 \text{ per day} \]

5. **Find the fraction left after 10 days:**
Using the decay formula again:
\[ \frac{N(10)}{N_0} = e^{-\lambda \cdot 10} = e^{-0.3465 \cdot 10} = e^{-3.465} \]

6. **Calculate \( e^{-3.465} \):**
Using a calculator or approximation:
\[ e^{-3.465} \approx 0.03125 \]
Therefore, the fraction left after 10 days is:
\[ \frac{N(10)}{N_0} \approx 0.03125 \]

7. **Calculate the fraction that decayed in 10 days:**
\[ \text{Fraction decayed} = 1 - \text{Fraction left} = 1 - 0.03125 = 0.96875 \]
Converting this to a fraction:
\[ 0.96875 = \frac{31}{32} \]

### Final Answer:
The fraction that decays in 10 days is \( \frac{31}{32} \).

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