Questions: A wire in the shape of a square with area 144 ft² is straightened and then reshaped into is rectangle with length √100 ft. Find the area of the rectangle.

A wire in the shape of a square with area 144 ft² is straightened and then reshaped into is rectangle with length √100 ft. Find the area of the rectangle.

Transcript text: 13. A wire in the shape of a square with area $144 \mathrm{ft}^{2}$ is straightened and then reshaped into is rectangle with length $\sqrt{100} \mathrm{ft}$. Find the area of the rectangle.

Solution

Solution Steps

To solve this problem, we need to follow these steps:

Calculate the side length of the square using its area.
Determine the perimeter of the square, which is the same as the length of the wire.
Use the given length of the rectangle to find its width by dividing the perimeter by 2 and subtracting the length.
Calculate the area of the rectangle using its length and width.

Step 1: Calculate the Side Length of the Square

Given the area of the square $ A = 144 \, \text{ft}^2 $, we can find the side length $ s $ using the formula: \[ s = \sqrt{A} = \sqrt{144} = 12 \, \text{ft} \]

Step 2: Calculate the Perimeter of the Square

The perimeter $ P $ of the square is calculated as: \[ P = 4s = 4 \times 12 = 48 \, \text{ft} \]

Step 3: Calculate the Width of the Rectangle

The length of the rectangle is given as $ l = \sqrt{100} = 10 \, \text{ft} $. The width $ w $ can be found using the relationship: \[ w = \frac{P}{2} - l = \frac{48}{2} - 10 = 24 - 10 = 14 \, \text{ft} \]

Step 4: Calculate the Area of the Rectangle

The area $ A_r $ of the rectangle is given by: \[ A_r = l \times w = 10 \times 14 = 140 \, \text{ft}^2 \]