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


_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

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

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.

_________________

Ankit

Check my Tutoring Site -> Brush My Quant

GMAT Quant Tutor
How to start GMAT preparations?
How to Improve Quant Score?
Gmatclub Topic Tags
Check out my GMAT debrief

How to Solve :
Statistics || Reflection of a line || Remainder Problems || Inequalities

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) / \(\pi\)