If the area of circle O is 16Ï€, what is the length

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BTGmoderatorLU: Moderator; Posts: 2505; Joined: Sun Oct 15, 2017 1:50 pm; Followed by:6 members

If the area of circle O is 16Ï€, what is the length

by BTGmoderatorLU » Tue May 15, 2018 3:10 pm

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If the area of circle O is 16Ï€, what is the length of an arc on the circle formed by a central angle measuring 45 degrees?

A. Ï€
B. 3Ï€/2
C. 2Ï€
D. 5Ï€/2
E. 8Ï€

The OA is A.

Please, can anyone assist me with this PS question? I don't have it clear. Thanks in advance.

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Source: Beat The GMAT — Problem Solving | Tue May 15, 2018 3:10 pm

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

by Vincen » Wed May 16, 2018 12:07 am

Hello BTGmoderatorLU.

Since the area of the circle O is 16Ï€, then we have that $$A=r^2\pi=16\pi\ \ \ \ \ \Leftrightarrow\ \ \ \ \ \ \ r^2=16\ \ \ \ \Leftrightarrow\ \ \ \ r=4.$$ Now, the length of an arc is equal to $$L=\theta\cdot r,\ \ \ where\ \theta\ is\ the\ central\ angle.$$ In our case, the central angle is 45Âº which is the same as Ï€/4 and the radius is r=4.

Hence we get $$L=\theta\cdot r=\frac{\pi}{4}\cdot4=\pi.$$ Therefore, the correct answer is the option [spoiler]A. Ï€[/spoiler].

I hope it can helps you.

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 am; Location: New York, NY; Thanked: 39 times; Followed by:22 members

by Jeff@TargetTestPrep » Thu May 17, 2018 5:21 pm

BTGmoderatorLU wrote:If the area of circle O is 16Ï€, what is the length of an arc on the circle formed by a central angle measuring 45 degrees?

A. Ï€
B. 3Ï€/2
C. 2Ï€
D. 5Ï€/2
E. 8Ï€

Let's first determine the radius of the circle. Since the area is 16Ï€:

Area = Ï€ x radius^2

16Ï€ = Ï€ x radius^2

16 = radius^2

radius = 4

Since circumference = 2Ï€(radius), the circumference = 2Ï€(4) = 8Ï€.

We need to determine the length of an arc on the circle formed by a central angle measuring 45 degrees. We recall that a full circle has 360 degrees.

We can create the following proportion to determine the length of the arc (which we denote as x):

45/360 = x/8Ï€

1/8 = x/8Ï€

8x = 8Ï€

x = Ï€

Answer: A

Jeffrey Miller
Head of GMAT Instruction
[email protected]