https://file.aiquickdraw.com/mathbot/file/1727478198924zh6h008d.jpg

The image you've uploaded appears to be a screenshot of a document containing mathematical questions about radioactive decay, using the decay function \( M(t) = M_0 \times 2.5^{-0.002t} \), where \( t \) is time in years.

The problems address:
1. Time taken for the substance to decay to 45% and 44% of its original amount and the difference between these times.
2. Time taken for the substance to decay to 10% of its original amount.

**Solution Approach for Part (a) and (b):**
- Use the decay formula to set up an equation based on the percentage remaining (either 45% or 10%) and solve for \( t \).
- For (c), the difference in times for the substance to decay from 45% to 44% is sought.

### Answer Verification and Explanation:

1. **For 45% Remaining:**
   \[
   0.45 = 2.5^{-0.002t}
   \]
   Taking the natural logarithm:
   \[
   \ln(0.45) = -0.002t \ln(2.5)
   \]
   Solving for \( t \):
   \[
   t = \frac{\ln(0.45)}{-0.002 \ln(2.5)}
   \]

2. **For 10% Remaining:**
   \[
   0.10 = 2.5^{-0.002t}
   \]
   Similarly, taking the natural logarithm and solving for \( t \):
   \[
   t = \frac{\ln(0.10)}{-0.002 \ln(2.5)}
   \]

### Part (c):
- The reason the answers for (a) and (b) are not the same is that the percentage decay is not linear over time. The exponential decay function models the process whereby the amount of substance decreases rapidly at first and then more slowly as time progresses.

Let's calculate the exact values for the times in years for each decay percentage and check the given solutions:
1. Time for the substance to decay to 45%.
2. Time for the substance to decay to 44%.
3. Time for the substance to decay to 10%.
4. Verify why the time differences for small percentage changes are significant as the substance approaches complete decay.

Let me perform the calculations.Here are the computed results based on the exponential decay model \( M(t) = M_0 \times 2.5^{-0.002t} \):

1. **Time to decay to 45% of original amount**: Approximately 435.73 years.
2. **Time to decay to 44% of original amount**: Approximately 447.99 years.
3. **Time to decay to 10% of original amount**: Approximately 1256.47 years.

### Time Differences:
- **From 45% to 44%**: The time difference is about 12.26 years.
- **From 45% to 10%**: The time difference is about 820.74 years.

### Explanation:
- The difference in time for the substance to decay from 45% to 44% (12.26 years) shows how sensitive the decay process is to small percentage changes

Suppose that a radioactive substance decays according to M(t) = 2.5e^(-0.002t), where t is time in years. How long will it take for the substance to reach certain percentages of its original amount?

Learn how to solve exponential decay problems with a step-by-step guide using logarithmic transformations. This problem explores the decay of a radioactive substance and calculates the time to reach specific percentages of the original amount. Suitable for advanced high school and university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation