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

Let's break down the formula Clare wrote for the area of the paper after \( n \) cuts:

\[ c(n) = 80 \times \left(\frac{1}{2}\right)^n \]

### Understanding the terms:
1. **80**: This is the initial area of the paper in square inches before any cuts are made. The area of a rectangle is given by multiplying its length and width:
   \[
   \text{Area} = \text{Length} \times \text{Width} = 8 \times 10 = 80 \text{ square inches}.
   \]
   So, the \( 80 \) represents the total area of the paper before any cuts.

2. **\( \left(\frac{1}{2}\right)^n \)**: Each time Clare cuts the paper in half, the area is reduced by half. After one cut, the area is halved, so the new area is \( \frac{1}{2} \) of the original. After two cuts, the area is halved again, becoming \( \frac{1}{4} \) of the original, and so on. Therefore, after \( n \) cuts, the area is reduced by a factor of \( \left(\frac{1}{2}\right)^n \), where \( n \) is the number of cuts made.

### The full formula:

- The \( 80 \) represents the original area.
- The \( \left(\frac{1}{2}\right)^n \) represents the fraction of the original area that remains after \( n \) cuts, where each cut halves the remaining area.

Thus, the expression \( c(n) = 80 \times \left(\frac{1}{2}\right)^n \) gives the area of the paper after \( n \) cuts.

Would you like further details on this explanation or have any questions?

Here are 5 related questions that might help:
1. How would the formula change if Clare cut the paper into thirds instead of halves?
2. After how many cuts would the area of the paper be less than 1 square inch?
3. Can you express the change in area using a recursive function?
4. How would the dimensions of the paper change after each cut?
5. How many times must Clare cut the paper in half to reduce its area to 5 square inches?

**Tip**: When solving problems with exponential decay (like halving), pay attention to how powers of fractions decrease the quantity.

Explore Clare's exponential decay formula for cutting paper in half repeatedly. The problem involves calculating the area of an 8x10 inch paper after multiple cuts using c(n) = 80 * (1/2)^n. Suitable for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation