Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again. . .

Instead of writing a recursive definition, Clare writes c(n)=80x(1/2)^n , where  is the area, in square inches, of the paper after  cuts. Explain where the different terms in her expression came from.

Clare's expression for the area of the paper after \(n\) cuts is given by \( c(n) = 80 \times \left(\frac{1}{2}\right)^n \), where \( c(n) \) represents the area of the paper after \(n\) cuts.

Let's break down the expression:

1. **Initial Area (80 square inches):**
   - The original piece of paper has dimensions 8 inches by 10 inches.
   - The area of the paper is calculated as \( 8 \times 10 = 80 \) square inches.
   - So, the term "80" in the equation refers to the **initial area** of the paper before any cuts are made.

2. **Factor of \( \left(\frac{1}{2}\right)^n \):**
   - Each time Clare cuts the paper in half, she reduces its area by half.
   - After one cut, the area is reduced to half of the original area: \( 80 \times \frac{1}{2} \).
   - After two cuts, the area is halved again: \( 80 \times \left(\frac{1}{2}\right)^2 \).
   - In general, after \(n\) cuts, the area becomes \( 80 \times \left(\frac{1}{2}\right)^n \), where \( \left(\frac{1}{2}\right)^n \) accounts for the area being halved \(n\) times.

### Explanation of Terms:
- **80**: Represents the starting area of the paper (8 inches by 10 inches).
- **\( \left(\frac{1}{2}\right)^n \)**: Represents the effect of cutting the paper in half \(n\) times. Each cut reduces the area by a factor of \( \frac{1}{2} \), so after \(n\) cuts, the area is reduced by \( \left(\frac{1}{2}\right)^n \).

### Summary:
The expression \( c(n) = 80 \times \left(\frac{1}{2}\right)^n \) is used to describe how the area of the paper decreases after each cut. The "80" is the initial area, and the \( \left(\frac{1}{2}\right)^n \) accounts for the fact that the area is halved \(n\) times.

Would you like more details or have any questions?

Here are five related questions:
1. How would the expression change if Clare started with a different sized paper?
2. How many cuts would it take for the area to be less than 1 square inch?
3. Can you derive a recursive formula for this problem?
4. What happens to the shape of the paper after multiple cuts?
5. How would the problem change if the paper were cut into thirds instead of halves?

**Tip**: When working with exponential decay problems like this, the base of the exponent (in this case \( \frac{1}{2} \)) represents the fraction by which the quantity decreases after each step.

Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again. . . Instead of writing a recursive definition, Clare writes c(n)=80x(1/2)^n, where is the area, in square inches, of the paper after cuts. Explain where the different terms in her expression came from.

Explore how cutting a piece of paper in half repeatedly leads to exponential decay in its area. This problem involves geometric and algebraic concepts, including calculating the initial area of a rectangle and using exponential decay formulas. Suitable for students in Grades 6-8.

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