https://file.aiquickdraw.com/mathbot/file/173030994611279tyz27x.jpg

Let's analyze the expressions for Eric and Andrea in the problem and break down each component.

### Eric's Expression: \( 10 \left( \frac{1}{4} \right)^w \)
1. **10**: This represents the **initial amount of the substance** that Eric starts with, which is 10 milligrams.
2. **\( \frac{1}{4} \)**: This represents the **rate of decay** per week, indicating that each week only one-fourth of the previous week's amount remains.
3. **\( w \)**: This is the **number of weeks** that have passed.
4. **\( \left( \frac{1}{4} \right)^w \)**: This represents the **remaining fraction** of the substance after \( w \) weeks, as the amount decreases by a factor of \( \frac{1}{4} \) each week.

### Andrea's Expression: \( (1 - 0.2)^w \)
1. **1**: Represents the **full amount** before decay is considered.
2. **0.2**: This represents the **decay rate per week** (20%).
3. **\( 1 - 0.2 \)**: Represents the **remaining portion** of the substance each week, meaning 80% (or 0.8) remains each week.
4. **\( w \)**: This is the **number of weeks**.

These drop-down explanations should help to clarify each part in terms of its role in modeling the exponential decay of the radioactive substances in Eric's and Andrea's cases.

Would you like a more detailed explanation or have questions on how exponential decay works?

---

Here are some related questions to explore:

1. What would the expressions look like if Eric’s decay rate was changed to one-third per week?
2. How would Andrea’s expression change if her substance decayed by 30% per week instead of 20%?
3. What is the difference in the decay rates of Eric’s and Andrea’s substances after 5 weeks?
4. Can you rewrite Andrea’s expression in terms of a base of 0.8 instead of \(1 - 0.2\)?
5. How can you determine the remaining substance if you know the decay rate but not the initial amount?

**Tip:** In exponential decay, the base of the exponential (like \( \frac{1}{4} \) or \(0.8\)) shows what fraction of the substance remains after each unit of time.

Eric starts with 10 milligrams of a radioactive substance. The amount of the substance decreases by one-fourth each week for a number of weeks, w. He writes the expression 10(1/4)^w to find the amount of radioactive substance remaining after w weeks. Andrea starts with 1 milligram of a radioactive substance. The amount of the substance decreases by 20% each week for a number of weeks, w. She writes the expression (1 - 0.2)^w to find the amount of radioactive substance remaining after w weeks.

Explore how exponential decay works with two examples: Eric's radioactive substance decreasing by one-fourth each week and Andrea's decreasing by 20% each week. Learn to analyze the expressions 10(1/4)^w and (1 - 0.2)^w to find the remaining amount after w weeks. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation