1 Answer

Bestsss · Answer 1 · 2023-07-29T15:12:58+0000

Step 1. Given:

-The square piece of paper that Kitan cut has a side length of 6 inches

-The sector of a circle that Mai cut has a radius of 6 inches and an arc length of 4pi.

Required: Find whose paper is larger.

We start by making a diagram of the circle sector and the square as shown in the image:

Step 2. To find which one is larger, we will need to find the area of the two figures.

First, we calculate the area of the square as follows:

Since the length is 6 in, the area is:

$\begin{gathered} A_(square)=(6in)^2 \\ \downarrow \\ \boxed{A_(square)=36in^2} \end{gathered}$

Step 3. To find the area of the circle sector, first, we have to find which part of a total circle the sector represents.

Let's find the circumference of the whole circle:

$\begin{gathered} C=2\pi r \\ C=2\pi(6in) \\ C=12\pi in \end{gathered}$

If we had a complete circle, the total arc or circumference would be 12pi:

But note that the sector is only 4pi, this means that the sector represents 1/3 of a complete circle.

Step 4. Once we know that the sector is one-third of a complete circle, we find its area by dividing the total area of the circle by 3:

$A_(circle)=(\pi r^2)/(3)$

Substituting the known values and using 3.14 as the value of pi:

$A_(c\imaginaryI rcle)=((3.14)(6in)^2)/(3)$

Solving the operations:

$\begin{gathered} A_(c\imaginaryI rcle)=((3.14)(36\imaginaryI n^2))/(3) \\ \downarrow \\ \boxed{A_{c\mathrm{i}rcle}=37.68in^2} \end{gathered}$

Step 5. Comparing the two areas:

$\begin{gathered} \boxed{A_(square)=36\imaginaryI n^(2)} \\ \boxed{A_{c\mathrm{\imaginaryI}rcle}=37.68\imaginaryI n^(2)} \end{gathered}$

We can see that the area of the circle sector is larger, therefore Mai's paper is larger.

Answer: Mai's paper is larger

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