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Probability and Geometry

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by tlt2372 » Tue Sep 28, 2010 11:17 am
This problem is making my head explode! Please say there is an easy way to do it!

Thanks!


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OA: E
Last edited by tlt2372 on Tue Sep 28, 2010 12:35 pm, edited 1 time in total.
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by goyalsau » Tue Sep 28, 2010 11:49 am
tlt2372 wrote:This problem is making my head explode! Please say there is an easy way to do it!

Thanks!


Image
If the answer is C then i can try to explain otherwise otherwise i am also in cue for the solution.
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by GMATGuruNY » Tue Sep 28, 2010 12:07 pm
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. root (2 (root 6))
b. (root 6 (root 6))/2
c. root (2 root 3)
d. root 3
e. 2

Let P = pi

circumference = 2Pr
2Pr = 4*√(P√3)
r = 4*√(P√3) / (2P)
= 2*√(P√3) / P

area = Pr^2
= P * [2*√(P√3) / P]^2
= P * 4P√3 / P^2
= 4√3

Since P(outside triangle) = 3/4, P(triangle) = 1/4

triangle = 1/4 * 4√3 = √3

area of equilateral triangle = 1/4 * b^2 *√3
1/4 * b^2 *√3 = √3
b^2 = 4
b=2

The correct answer is E.
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by goyalsau » Tue Sep 28, 2010 12:14 pm
area of equilateral triangle = 1/4 * b^2 *√3

Only formula i know for the area of a triangle is = (Base * Height )/2

Can you please explain how to reach to that formula that you mentioned above.
Because as i feel its easy to drive formula's in comparison remember them.

thanks.
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by GMATGuruNY » Tue Sep 28, 2010 12:33 pm
goyalsau wrote:area of equilateral triangle = 1/4 * b^2 *√3

Only formula i know for the area of a triangle is = (Base * Height )/2

Can you please explain how to reach to that formula that you mentioned above.
Because as i feel its easy to drive formula's in comparison remember them.

thanks.
The attached drawing shows how the formula for the area of an equilateral triangle is derived.
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equilateral_area.pdf
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by goyalsau » Tue Sep 28, 2010 2:04 pm
GMATGuruNY wrote:
goyalsau wrote:area of equilateral triangle = 1/4 * b^2 *√3

Only formula i know for the area of a triangle is = (Base * Height )/2

Can you please explain how to reach to that formula that you mentioned above.
Because as i feel its easy to drive formula's in comparison remember them.

thanks.
The attached drawing shows how the formula for the area of an equilateral triangle is derived.
Thanks guru......
Saurabh Goyal
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by diebeatsthegmat » Tue Sep 28, 2010 4:44 pm
GMATGuruNY wrote: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. root (2 (root 6))
b. (root 6 (root 6))/2
c. root (2 root 3)
d. root 3
e. 2

Let P = pi

circumference = 2Pr
2Pr = 4*√(P√3)
r = 4*√(P√3) / (2P)
= 2*√(P√3) / P

area = Pr^2
= P * [2*√(P√3) / P]^2
= P * 4P√3 / P^2
= 4√3

Since P(outside triangle) = 3/4, P(triangle) = 1/4

triangle = 1/4 * 4√3 = √3

area of equilateral triangle = 1/4 * b^2 *√3
1/4 * b^2 *√3 = √3
b^2 = 4
b=2

The correct answer is E.
thanks a lot,
however i dont understand the circumsference part.
i was thinking of the circumference of cyn linrical tank then got confused...
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by GMATGuruNY » Wed Sep 29, 2010 8:38 am
diebeatsthegmat wrote:
GMATGuruNY wrote: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. root (2 (root 6))
b. (root 6 (root 6))/2
c. root (2 root 3)
d. root 3
e. 2

Let P = pi

circumference = 2Pr
2Pr = 4*√(P√3)
r = 4*√(P√3) / (2P)
= 2*√(P√3) / P

area = Pr^2
= P * [2*√(P√3) / P]^2
= P * 4P√3 / P^2
= 4√3

Since P(outside triangle) = 3/4, P(triangle) = 1/4

triangle = 1/4 * 4√3 = √3

area of equilateral triangle = 1/4 * b^2 *√3
1/4 * b^2 *√3 = √3
b^2 = 4
b=2

The correct answer is E.
thanks a lot,
however i dont understand the circumsference part.
i was thinking of the circumference of cyn linrical tank then got confused...
The problem says that the circumference of the base -- which is a circle = 4*√(P√3).
The formula for the circumference of a circle = 2Pr.
So 2Pr = 4*√(P√3)

Divide each side by 2P:
r = 4*√(P√3) / (2P)

Divide the 4 in the numerator by the 2 in the denominator:
r = 2*√(P√3) / P
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My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
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