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In the circle above, if the area of the rectangle set inside the circl

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05 Jul 2018, 04:52
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35% (medium)

Question Stats:

71% (02:42) correct 29% (02:53) wrong based on 31 sessions

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In the circle above, if the area of the rectangle set inside the circle is 200 and b = 8a, what is the circumference of the circle?

A. $$25\pi \sqrt{65}$$

B. $$5 \sqrt{65}$$

C. $$5\pi \sqrt{13}$$

D. $$5\pi \sqrt{65}$$

E. $$5\pi$$

Attachment:

circle.png [ 7.74 KiB | Viewed 384 times ]

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05 Jul 2018, 05:02
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Given: b = $$8a$$(where $$a$$ is the width of the rectangle) | Area(rectangle) = 200

The product of length and breadth is the area of the rectangle -> $$a*8a = 200$$

Solving for a, we will get $$a^2 = \frac{200}{8} = 25$$ -> $$a = \sqrt{25} = 5$$.

From the figure, we know that the length of the diagonal is the diameter of the circle.

Length of diagonal = $$\sqrt{a^2 + (8a)^2} = a\sqrt{65} = 5\sqrt{65}$$

Therefore, the circumference of the circle is $$\pi * d = 5\pi \sqrt{65}$$ (Option D)
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In the circle above, if the area of the rectangle set inside the circl  [#permalink]

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05 Jul 2018, 18:36
Bunuel wrote:

In the circle above, if the area of the rectangle set inside the circle is 200 and b = 8a, what is the circumference of the circle?

A. $$25\pi \sqrt{65}$$

B. $$5 \sqrt{65}$$

C. $$5\pi \sqrt{13}$$

D. $$5\pi \sqrt{65}$$

E. $$5\pi$$

Attachment:
circle.png

(1) Use the area of the rectangle to find side lengths, in which $$L=b=8a$$
$$A=(W*L)=(a*b)=(a*8a)=200$$
$$8a^2=200$$
$$a^2=25$$
$$a=5$$
$$b=8a=40$$

(2) Use Pythagorean theorem to find hypotenuse = diameter of circle*
$$a^2+b^2=d^2$$
$$5^2+40^2=d^2$$
$$d^2=1,625$$
Look at the answer choices. $$\sqrt{65}$$ shows up three times
$$1,625=65*25$$, so
$$\sqrt{d^2}=\sqrt{25*65}$$
$$d=5\sqrt{65}$$

(3) Circumference= $$\pi* d=5\sqrt{65}\pi$$

*Use variables instead of values to avoid 1,625
$$a^2+8a^2=d^2$$
$$a^2*(1+8^2)=d^2$$
$$\sqrt{a^2}*\sqrt{65}=\sqrt{d^2}$$
$$a*\sqrt{65}=d$$
$$d=5*\sqrt{65}$$

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