Probability and Geometry

tlt2372 wrote:This problem is making my head explode! Please say there is an easy way to do it!

Thanks!

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.

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.

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.

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...