GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 22 Oct 2019, 19:27

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# ABCD is a square inscribed in a circle and arc ADC has a length of

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58427
ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

28 Oct 2014, 09:35
2
14
00:00

Difficulty:

55% (hard)

Question Stats:

66% (02:59) correct 34% (02:45) wrong based on 201 sessions

### HideShow timer Statistics

ABCD is a square inscribed in a circle and arc ADC has a length of $$\pi\sqrt{x}$$. If a dart is thrown and lands somewhere in the circle, what is the probability that it will not fall within the inscribed square? (Assume that the point in the circle where the dart lands is completely random.)

(A) $$2x$$

(B) $$π(x) - 2x$$

(C) $$π(x) - \sqrt{2}(x)$$

(D) $$1 - \frac{2}{π}$$

(E) $$1 - \frac{2}{x}$$

Attachment:

2014-10-28_2033.png [ 10.59 KiB | Viewed 10492 times ]

_________________
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1747
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

Updated on: 08 Dec 2014, 03:47
12
1
Perimeter of the circle $$= 2 * \pi\sqrt{x}$$

So, radius of circle $$= \sqrt{x}$$

Diameter of the circle = Diagonal of the square $$= 2\sqrt{x}$$

Area of circle $$= \pi (\sqrt{x})^2 = \pi x$$

Area of square $$= \frac{(2\sqrt{x})^2}{2} = 2x$$

Area of the yellow shaded region $$= \pi x - 2x$$
Attachment:

rhom.png [ 11.37 KiB | Viewed 9781 times ]

Probability of NOT falling in inscribed square = Probability of FALLING in the shaded region

Probability $$= \frac{\pi x - 2x}{\pi x} = 1 - \frac{2}{\pi}$$

Bunuel: Kindly update the OA
_________________
Kindly press "+1 Kudos" to appreciate

Originally posted by PareshGmat on 28 Oct 2014, 21:41.
Last edited by PareshGmat on 08 Dec 2014, 03:47, edited 2 times in total.
##### General Discussion
Intern
Status: Preparing
Joined: 09 Oct 2014
Posts: 10
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

28 Oct 2014, 10:36
3
3
Length of the arc = pi*sqrt x
Radius of the circle = 180/360*2pi*r = pi*sqrt x ==> r = sqrt x
Area of the circle = pi*r^2 ==> pi*x

Diagonal of the square = diameter of the circle = 2* sqrt x
Side(S) of the square =Diagonal/sqrt 2
==> S * sqrt 2 = 2*sqrt x ==> s=sqrt 2x
Area of the square = s^2 = 2x

Probability of the dart not landing on the square = 1-2x/pi*x ==> 1- 2/pi

Intern
Joined: 06 Jun 2011
Posts: 20
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

28 Oct 2014, 11:56
Circumference = 2 * length of arc ADC = 2 * pi * sqrt(x)
radius of circle = sqrt (x)

Probability = [area(circle) - area(square)] / area(circle)
= (pi - 2 )/ pi
GMAT Tutor
Status: Tutor - BrushMyQuant
Joined: 05 Apr 2011
Posts: 622
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE: Information Technology (Computer Software)
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

06 Dec 2014, 23:39
2
Hi Paresh,

Area of yellow shaded region is $$= \pi x - 2x$$ (circle - square) and not $$= 2x - \pi x$$ (square - circle)

kindly update the remaining steps too
Thank you.
PareshGmat wrote:
Area of the yellow shaded region $$= 2x - \pi x$$

_________________
Intern
Joined: 08 Jul 2012
Posts: 46
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

07 Dec 2014, 22:23
1
PareshGmat wrote:
nktdotgupta wrote:
Hi Paresh,

Area of yellow shaded region is $$= \pi x - 2x$$ (circle - square) and not $$= 2x - \pi x$$ (square - circle)

kindly update the remaining steps too
Thank you.
PareshGmat wrote:
Area of the yellow shaded region $$= 2x - \pi x$$

Its been updated. Thanks

Is there any way to update OA for this question?

Hi Paresh,

I believe the denominator of the probability should be area of Circle and not that of square.

So, the prob (of not falling in inscribed square) = 1 - P(falling in inscribed square) = 1 - (2*x/pi*x) = 1-2/pi

OR as per your approach prob (of not falling in inscribed square) = (pi*x - 2x) / pi*x = 1-2/pi
_________________
Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time. - Thomas A. Edison
Manager
Joined: 18 Jan 2010
Posts: 246
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

27 May 2016, 06:37
1
First Point Arc ADC is nothing but semi circle. Because AC is diameter.

so length of arc ADC is ($$\pi$$ r)

Given. ($$\pi$$ r = ($$\pi$$ $$\sqrt{x}$$

r = $$\sqrt{x}$$

Area of the Circle: $$\pi$$ $$r^2$$ = $$\pi$$x

Now let us calculate area of square.

Diagonal of square = 2r = 2$$\sqrt{x}$$

a $$\sqrt{2}$$ = 2$$\sqrt{x}$$; where a is the side of square.

a = $$\sqrt{2x}$$

Area of square = 2x.

We are asked probability that dart will not fall within the inscribed square?

Probability = ($$\pi$$x - 2x) / $$\pi$$x = ($$\pi$$ - 2) / $$\pi$$
Intern
Joined: 03 Dec 2017
Posts: 5
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

25 Jan 2018, 11:45
A very simple way to solve this question is by looking the alternatives. The probability of the dard fall outside the square does not depend on the size of the circle and the square, thus it does not depend on x. The only alternative that does not depend on x is letter d.
Non-Human User
Joined: 09 Sep 2013
Posts: 13412
Re: ABCD is a square inscribed in a circle and arc ADC has a length of  [#permalink]

### Show Tags

16 Aug 2019, 05:06
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: ABCD is a square inscribed in a circle and arc ADC has a length of   [#permalink] 16 Aug 2019, 05:06
Display posts from previous: Sort by