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Medium|   Geometry|      
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Donnie84
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Square ABCD has a side length of 4. BC is the diameter of the circle. Updated on: 12 Apr 2024, 16:24
Originally posted by Donnie84 on 20 Mar 2024, 13:22.
Last edited by Carcass on 12 Apr 2024, 16:24, edited 1 time in total.
Fixed the image and updated the question
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­Square ABCD has a side length of 4. BC is the diameter of the circle. Which of the following is greater than or equal to the area of the shaded region, in square units?

Indicate all possible choices.

A. \(16 − 16\pi\) 

B. \(16 − 4\pi\) 

C. \(16 − 2\pi\) 

D. \(16 + \pi\)

E. \(16 + 4\pi\)

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C,D,E
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Square ABCD has a side length of 4. BC is the diameter of the circle. 12 Apr 2024, 16:25
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­Area of the square ABCD , \(4^2 = 16\)

Area of ARC BC , 
\(\frac{180}{360}*(\frac{1}{4}*pi*d^2)\) [d= diameter= 4]
\(\frac{180}{360}*(\frac{1}{4}*pi*16)\)
\(2*pi\)

So the shaded area must be equal or greater than  \(16-2*pi\)

So the possible answers are C, D, E
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Square ABCD has a side length of 4. BC is the diameter of the circle. 12 Apr 2024, 16:25
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­Alternative explanation

\(AB =\) side of a square \(= 4\) = diameter

Radius \(= 2\)

Area of a semicircle with radius \(2 = \frac{πr^2}{2}\)

Area \(= 2π\)

Area of a square \(= 16\)

Area of a shaded region \(= 16 - 2π\)

We need the area that can be greater than or equal to the area of the shaded region.

Answer C D E
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Square ABCD has a side length of 4. BC is the diameter of the circle. 23 Jun 2024, 12:05
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Official Explanation

­The area of the shaded region is the area of the square minus the area of the portion of the circle that is inside the square. The area of a square is its side squared. The area of square ABCD is \(4^2 = 4 × 4\), which is 16. Now find the area of the portion of the circle that is inside the square. Because the diameter of the circle is a side of the square, you know that exactly one-half of the circle’s area is inside the square. Also, because the diameter of the circle is twice the radius, the radius of the circle is \(\frac{4}{2}\) or 2. The area of a circle with a radius r is \(πr^2\). The area of the complete circle in this question is \(π(2)^2\), which is \(4π\). So half the area of this circle is \(2π\). Thus, the area of the shaded region is \(16 - 2π\). That means that \(16 - 4π\) and 16 - 16π are less than \(16 - 2π\), so they cannot be correct choices. However, the sum of 16 and any positive number is greater than 16 and also greater than \(16 - 2π\). So, the correct choices are (C), (D), and (E).