How long will it take before fourteen percent of our 1,000

-gram sample of uranium-235 has decayed?

 

Round your answer to the nearest year.

To solve this problem, we need to determine the time it takes for 14% of the uranium-235 sample to decay. This involves using the concept of exponential decay and the half-life of uranium-235.

### Step-by-step Solution:
1. **Given:**
   - Initial amount of the sample, \( M_0 = 1000 \) grams.
   - Decayed amount, \( 14\% \), so \( 0.14 \times 1000 = 140 \) grams.
   - Half-life of uranium-235 is approximately 703.8 million years.

2. **Exponential Decay Formula:**
   The remaining amount after time \( t \) is given by:
   \[
   M(t) = M_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
   \]
   where \( T_{1/2} \) is the half-life.

3. **Determine the Remaining Mass after Decay:**
   If 140 grams has decayed, the remaining mass \( M(t) \) is:
   \[
   M(t) = 1000 - 140 = 860 \text{ grams}
   \]

4. **Set Up the Equation:**
   \[
   860 = 1000 \times \left(\frac{1}{2}\right)^{\frac{t}{703.8}}
   \]

5. **Solve for \( t \):**
   \[
   \frac{860}{1000} = \left(\frac{1}{2}\right)^{\frac{t}{703.8}}
   \]

   Simplify the ratio:
   \[
   0.86 = \left(\frac{1}{2}\right)^{\frac{t}{703.8}}
   \]

6. **Take the Natural Logarithm:**
   \[
   \ln(0.86) = \frac{t}{703.8} \ln\left(\frac{1}{2}\right)
   \]

7. **Solve for \( t \):**
   \[
   t = \frac{\ln(0.86)}{\ln(0.5)} \times 703.8
   \]

   Plug in the values:
   \[
   t \approx \frac{-0.15082}{-0.6931} \times 703.8
   \]

   \[
   t \approx 153.0 \text{ years}
   \]

### Final Answer:
It will take approximately **153 years** for 14% of the 1,000-gram sample of uranium-235 to decay.

Would you like further details on exponential decay or any other related questions?

How long will it take before fourteen percent of our 1,000-gram sample of uranium-235 has decayed? Round your answer to the nearest year.

Find out how to calculate the time required for 14% decay of a 1,000-gram sample of uranium-235. This detailed solution uses exponential decay and the half-life of uranium-235 to solve for time, ideal for Grades 10-12.

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