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Four circles, each with a radius of 2 inches, are removed from a square. What is the remaining area of the square?

A. [tex]\( (16-4\pi) \)[/tex] in.[tex]\(^2\)[/tex]
B. [tex]\( (16-\pi) \)[/tex] in.[tex]\(^2\)[/tex]
C. [tex]\( (64-16\pi) \)[/tex] in.[tex]\(^2\)[/tex]
D. [tex]\( (64-4\pi) \)[/tex] in.[tex]\(^2\)[/tex]

Sagot :

GinnyAnswer
Let's solve the problem step by step to find the remaining area of the square after removing four circles, each with a radius of 2 inches.

### Step 1: Calculate the Area of One Circle
The radius of each circle is given as 2 inches. The formula to find the area of a circle is:
[tex]\[ \text{Area of a Circle} = \pi \times \text{radius}^2 \][/tex]

Substituting the radius:
[tex]\[ \text{Area of one circle} = \pi \times 2^2 = 4\pi \text{ square inches} \][/tex]

### Step 2: Calculate the Total Area of the Four Circles
Since there are four circles, we multiply the area of one circle by 4:
[tex]\[ \text{Total Area of Four Circles} = 4 \times 4\pi = 16\pi \text{ square inches} \][/tex]

### Step 3: Calculate the Side Length of the Square
Each radius of the circle is 2 inches. Since we are given that four circles are removed from the square, it implies the side length of the square is the diameter of one circle multiplied by 2 (because we need to fit two circles side by side along one side of the square). The diameter of one circle is:
[tex]\[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 2 = 4 \text{ inches} \][/tex]

Therefore, the side length of the square is the same as the diameter of one circle since four circles make a cross pattern in the square with length:
[tex]\[ \text{Side Length of the Square} = 2 \times \text{Diameter} = 2 \times 2 \times 2 = 4 \text{ inches} \][/tex]

### Step 4: Calculate the Area of the Square
To find the area of the square, use the side length:
[tex]\[ \text{Area of the Square} = \text{Side Length}^2 = 4^2 = 16 \text{ square inches} \][/tex]

### Step 5: Calculate the Remaining Area
The remaining area of the square after removing the area occupied by the four circles is:
[tex]\[ \text{Remaining Area} = \text{Area of the Square} - \text{Total Area of Four Circles} \][/tex]

Substitute the values:
[tex]\[ \text{Remaining Area} = 16 - 16\pi \text{ square inches} \][/tex]

So, the final solution provided by this is:
[tex]\[ \boxed{16 - 16\pi \text{ square inches}} \][/tex]

Hence, the correct answer is:
[tex]\[ (16 - 16\pi) \text{ square inches} \][/tex]
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