Bunuel wrote:

In the figure above, if semicircles A and B each have area \(4\pi\), what is the area of semicircle C?

(A) \(4\pi\)

(B) \(4\pi \sqrt{2}\)

(C) \(6\pi\)

(D) \(8\pi\)

(E) 16

Attachment:

2018-03-16_1011_003.png

Since both semi-circles A and B have the same area \(4*\pi\)

they will have similar radii(r), which can be calculated as follows:

\(\frac{1}{2}*\pi*r^2 = 4*\pi\) -> \(r^2 = 8\) ->\(r = 2\sqrt{2}\)

The diameter of the two semicircle \(2r = 4\sqrt{2}\) are the equal sides of an isosceles right-angled triangle.

The sides of these type of triangle are in the ratio \(1:1:\sqrt{2}\).

The hypotenuse of this right-angled triangle is the diameter of the semicircle C, which is \(\sqrt{2} * 4\sqrt{2} = 8\)

Therefore, the area of the semicircle C is \(\frac{1}{2}*\pi*r^2 = \frac{1}{2}*\pi*4^2 = 8*\pi\)(Option D)

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