https://file.aiquickdraw.com/mathbot/file/17267914470111sjldjko.jpg

Let's address the two questions from the image:

### Question 9:
"Write the exponential equation that MODELS this situation."

- Based on the format requirements, the general form for an exponential equation is:
  \[
  y = a(1 + r)^t
  \]
  where:
  - \( a \) is the initial amount,
  - \( r \) is the rate of growth or decay (as a decimal),
  - \( t \) is the time.

For decay situations (like radioactive decay), \( r \) will be negative.

### Question 10:
"How many grams of Pu-241 are left after 43.2 years?"

- For radioactive decay, we typically use the formula:
  \[
  A = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{h}}
  \]
  where:
  - \( A_0 \) is the initial amount,
  - \( A \) is the remaining amount,
  - \( t \) is the time,
  - \( h \) is the half-life of the substance.

If the half-life of Pu-241 is provided or known, we can calculate the remaining amount after 43.2 years.

**Please provide the initial amount and half-life of Pu-241 for precise calculation, or if you need clarification on any part.**

Would you like more details or have any specific questions?

### Related Questions:
1. What is the formula for exponential growth and how does it differ from decay?
2. How do you determine the growth or decay rate \( r \) from a percentage?
3. How does the half-life formula relate to the exponential decay equation?
4. What are some real-life applications of exponential growth and decay?
5. How can you derive the half-life if the decay constant \( \lambda \) is known?

**Tip:** When solving exponential problems, always verify whether the situation involves growth (positive \( r \)) or decay (negative \( r \)).

Write the exponential equation that MODELS this situation. How many grams of Pu-241 are left after 43.2 years?

Learn how to model exponential decay and calculate the remaining grams of Pu-241 after 43.2 years. This problem demonstrates key concepts in exponential functions and radioactive decay, suitable for students in Grades 9-12.

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