Bunuel wrote:
A circle is inscribed in a square ABCD such that the circle touches all the sides of the square. If the perimeter of the shaded region is 24 + 6π, what is the area of the shaded region?
A. 16 - 4π
B. 32 - 8π
C. 48 - 12π
D. 64 - 16π
E. 48 - 8π
Breaking Down the Info:> First, we can "complete" the perimeter by accounting for the entire shape. We are given the perimeter for \(\frac{3}{4}\) of the shape only, so multiply \(24 + 6\pi\) by \(\frac{4}{3}\) to get \(32 + 8\pi\).
> \(32 + 8\pi\) is the perimeter of the square + the circumference of the circle. Typically we can guess 32 belongs to the perimeter of the square and \(8\pi\) belongs to the circumference of the circle. Check that this gives a diameter of 8 and a length of 8 from the square; this confirms our guess was correct.
> Finally the shaded region is \(\frac{3}{4}\) of the difference of the area of the square and the area of the circle. Hence \(\frac{3}{4}*(8^2 - 4^2\pi) = 48 - 12\pi\).
Answer: C _________________
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