Two overlapping circles with equal radius form a shaded region as shown in the figure. Express the area of the region and the complete perimeter (combined arc length) in terms of r and the distance between center, #D#? Let #r=4 and D=6# and calculate?

Answer 1

see explanation.

Given #AB=D=6, => AG=D/2=3#
Given #r=3#
#=> h=sqrt(r^2-(D/2)^2)=sqrt(16-9)=sqrt7#
#sinx =h/r=sqrt7/4#
#=> x=41.41^@#

Area GEF (red area) #=pir^2*(41.41/360)-1/2*3*sqrt7#
#=pi*4^2*(41.41/360)-1/2*3*sqrt7=1.8133#

Yellow Area # = 4 * #Red Area #= 4*1.8133=7.2532#

arc perimeter #(C->E->C)=4xx2pirxx(41.41/360)#
#= 4xx2pixx4xx(41.41/360)=11.5638#

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Answer 2

The area of the shaded region is (2\pi r^2 - 2\left(\frac{1}{2} \pi r^2\right) = \pi r^2). The perimeter of the shaded region is (2\pi r + 2\pi \left(r\cos^{-1}\left(\frac{D}{2r}\right)\right)).

Given (r=4) and (D=6),

The area of the shaded region is (\pi \times 4^2 = 16\pi) square units.

The perimeter of the shaded region is (2\pi \times 4 + 2\pi \left(4\cos^{-1}\left(\frac{6}{2\times 4}\right)\right) = 8\pi + 2\pi \left(4\cos^{-1}\left(\frac{3}{4}\right)\right)). Using trigonometric identities, (\cos^{-1}\left(\frac{3}{4}\right) = \frac{\pi}{5}), So, the perimeter is (8\pi + 2\pi \times 4 \times \frac{\pi}{5} = 8\pi + \frac{8\pi}{5} = \frac{48\pi}{5}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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