Hi bhushan288, and welcome to Gmat Club.

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Original question is:

A cylindrical tank has a base with a circumference of 4\sqrt{\pi{\sqrt{3}} meters and an equilateral triangle painted on the interior side of the base. A grain of sand is dropped into the tank, and has an equal probability of landing on any particular point on the base. If the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4, what is the length of a side of the triangle?

A. \sqrt{2{\sqrt{6}}

B. \frac{\sqrt{6{\sqrt{6}}}}{2}

C. \sqrt{2{\sqrt{3}}

D. \sqrt{3}

E. 2

Given: circumference=4\sqrt{\pi{\sqrt{3}} and P(out)=\frac{3}{4}

Now, as the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4 then the the portion of the base (circle) outside the triangle must be 3/4 of the are of the base and the triangle itself 1/4 of the are of the base.

Next: circumference=4\sqrt{\pi{\sqrt{3}}}=2\pi{r} --> square both sides --> 16\pi{\sqrt{3}}=4{\pi}^2{r}^2 --> 4{\sqrt{3}}={\pi}{r}^2 --> area_{base}=\pi{r^2}=4{\sqrt{3}};

The area of the equilateral triangle is 1/4 of the base: area_{equilateral}=\frac{1}{4}*4{\sqrt{3}}=\sqrt{3} --> also the ares of the equilateral triangle is area_{equilateral}=a^2*\frac{\sqrt{3}}{4}, where a is the length of a side --> area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\sqrt{3} --> a=2.

Answer: E.
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A cylindrical tank has a base with a circumference of 4(sqrt(pi sqrt(3)) meters and an equilateral triangle painted on the interior side of the base. A grain of sand is dropped into the tank, and has an equal probability of landing on any particular point on the base. If the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4, what is the length of a side of the triangle?

A. \sqrt{2{\sqrt{6}}

B. \frac{\sqrt{6{\sqrt{6}}}}{2}

C. \sqrt{2{\sqrt{3}}

D. \sqrt{3}

E. 2

Given: circumference=4\sqrt{\pi{\sqrt{3}} and P(out)=\frac{3}{4}

Now, as the probability of the grain of sand landing on the portion of the base outside the triangle is 3/4 then the the portion of the base (circle) outside the triangle must be 3/4 of the are of the base and the triangle itself 1/4 of the are of the base.

Next: circumference=4\sqrt{\pi{\sqrt{3}}}=2\pi{r} --> square both sides --> 16\pi{\sqrt{3}}=4{\pi}^2{r}^2 --> 4{\sqrt{3}}={\pi}{r}^2 --> area_{base}=\pi{r^2}=4{\sqrt{3}};

The area of the equilateral triangle is 1/4 of the base: area_{equilateral}=\frac{1}{4}*4{\sqrt{3}}=\sqrt{3} --> also the ares of the equilateral triangle is area_{equilateral}=a^2*\frac{\sqrt{3}}{4}, where a is the length of a side --> area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\sqrt{3} --> a=2.

Answer: E.[/quote]

Hello Bunuel

ur explanation is perfect . i just cant understand one thing. i know that formula for the side of equilateral triangle inscribed in the circle
should be a= √3 * r where r is radius and a is side of the triangle, but when using this formula i am not getting the right answer in the above exmpl. what could be the problem? is somth. wrong with formula ? thanks

2*pi*r = 4(sqrt(pi sqrt(3))

=> r = 4(sqrt(pi sqrt(3))/2*pi = 2(sqrt(pi sqrt(3))/pi

Area of circle C = pi * r^2 = 4 * pi* 1/(pi)^2 * pi * sqrt(3) = 4*sqrt(3)


A/C = 1/4

=> A = sqrt(3)

Area of Triangle = sqrt(3) = sqrt(3)/4 * (side)^2

So side = sqrt(4) = 2

Answer is E.
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Because we are told that equilateral triangle is painted so not necessarily inscribed on the interior side of the base.
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Bigger the area bigger the probability of a grain landing there. P(out)=3/4 simply means that the the portion of the base (circle) outside the triangle must be 3/4 of the are of the base.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders

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. NEW!!! ,11 Mixed Questions NEW!!!, 12 Fresh Meat NEW!!!

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. NEW!!!, 11 New DS set. NEW!!!


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a=2. Check here: a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html#p824188
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders

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. NEW!!! ,11 Mixed Questions NEW!!!, 12 Fresh Meat NEW!!!

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. NEW!!!, 11 New DS set. NEW!!!


What are GMAT Club Tests?
25 extra-hard Quant Tests

Find out what's new at GMAT Club - latest features and updates