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A square with a side length of 6 feet has a circle with the largest possible radius cut out from it. What will be the approximate area, in square feet, of the remaining board?

[tex]\[ A = \pi r^2 \text{ where } \pi = 3.14 \][/tex]

A. [tex]\( 8 \, \text{ft}^2 \)[/tex]

B. [tex]\( 16 \, \text{ft}^2 \)[/tex]

C. [tex]\( 24 \, \text{ft}^2 \)[/tex]

D. [tex]\( 36 \, \text{ft}^2 \)[/tex]



Answer :

GinnyAnswer
Let's solve the problem step by step.

1. Calculate the area of the square:
The side length of the square is given as 6 feet.

Area of the square [tex]\( A_{\text{square}} \)[/tex] is calculated using the formula:
[tex]\[ A_{\text{square}} = \text{side length}^2 = 6^2 = 36 \text{ square feet} \][/tex]

2. Determine the diameter and radius of the largest possible circle:
The diameter of the largest possible circle that can be cut out of the square will be equal to the side length of the square, which is 6 feet.

So, the diameter [tex]\( d \)[/tex] of the circle is 6 feet.

The radius [tex]\( r \)[/tex] of the circle is half of the diameter:
[tex]\[ r = \frac{d}{2} = \frac{6}{2} = 3 \text{ feet} \][/tex]

3. Calculate the area of the circle:
The area of the circle [tex]\( A_{\text{circle}} \)[/tex] is calculated using the formula:
[tex]\[ A_{\text{circle}} = \pi r^2 \][/tex]
Given [tex]\(\pi = 3.14\)[/tex] and [tex]\(r = 3\)[/tex] feet, we have:
[tex]\[ A_{\text{circle}} = 3.14 \times 3^2 = 3.14 \times 9 = 28.26 \text{ square feet} \][/tex]

4. Calculate the area of the remaining board:
The area of the remaining board is the difference between the area of the square and the area of the circle:
[tex]\[ A_{\text{remaining}} = A_{\text{square}} - A_{\text{circle}} = 36 - 28.26 = 7.74 \text{ square feet} \][/tex]

Therefore, the approximate area of the remaining board after cutting out the largest possible circle is [tex]\(\boxed{7.74}\)[/tex] square feet.

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