Probability Triangle

[Annie]

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

Answer is e.

I understand the whole explanation except for "here the triangle has an area of root 3"

[StaceyKoprince]

Can you double check the problem? You've typed above "has a base with a circumference of meters..." How many meters? (Note: if it included symbols, and you just copied and pasted, it won't come through properly - you have to type it in yourself.)

I can guess, though that it might have something to do with comparing the area of the triangle to the area of the circular base. If the sand has a 3/4 probability of landing outside of the triangle, then it also has a 1/4 probability of landing inside the triangle. If we say the total area of the circle = area of triangle + area of circle OUTSIDE of triangle, then the 1/4 probability corresponds to the triangle's area and the 3/4 probability corresponds to the area outside the triangle. If I can calculate the area of hte circle, I can calculate the area of the triangle (that is, 1/4 the area of the circle).

[matt.mcmahon]

It is 4(root(pi(root 3))) meters

[RonPurewal]

It is 4(root(pi(root 3))) meters

if that's the circumference, then the radius is this quantity divided by 2p. (here 'p' stands for pi)
which is
(4√(P√3)) / 2P
= 2√(P√3)) / P **
= 2√√3 / √P *** - if you don't understand this step, i'll also show the work starting from (**).

starting from (***):
circle area = P(r^2)
= P * 4√3/P
= 4√3
so triangle area = 1/4 of this = √3

starting from (**):
circle area = P(r^2)
= P * 4P√3 / P^2
= 4√3
so triangle area = 1/4 of this = √3

[blaad]

How will you find out the length of a side of an equilateral triangle with area root 3?

[shoboy]

area of a triangle = 1/2*base*height. If one side of equilateral triangle = a, then to get the height, bissect the triangle and use pythatgoras theory to solve for the height, i.e height = root(a squared - a squared/4) = root(3a squared/4) =a(root3)/2. We can then solve for "a" :- 1/2*base*height = root3, where base =a, and height =a(root3)/4, so 1/2*a*a(root3)/2 = root3. If you solve the equation, you get a = 2.

[rfernandez]

shoboy correctly derives the formula for the area of an equilateral triangle in terms of its side.

Alternatively, I have found it helpful to memorize this formula: A = s^2 * root(3) / 4, where s is the length of the triangle's side.

If A is given as root(3), then:

root(3) = s^2 * root(3) / 4
1 = s^2 / 4
4 = s^2
2 = s

I know it's yet another formula to memorize, but it comes in handy.

[vrajesh.dave]

Here is the explaination from the MGMAT Cat


I understand that the area of triangle = 1/2 * base * height
hence Area of equilateral triangle can be express as 1/2 * S * (S *root(3))/2

But how do we get area of triangle as root(3).

[Ben Ku]

Here is the explaination from the MGMAT Cat


I understand that the area of triangle = 1/2 * base * height
hence Area of equilateral triangle can be express as 1/2 * S * (S *root(3))/2

But how do we get area of triangle as root(3).

In the explanation, the area of the circular base was found to be 4 sqrt(3). Because the equilateral triangle is 1/4 of the area of the base, therefore it has an area of sqrt(3).

Hope that helps.

[vjsharma25]

I approached it differently and was getting somewhat different result.
You don't need the probability statement at all for solving this question.
You can calculate the radius as the circumference of the base is given
radius = (4√(P√3)) / 2P
= 2√(P√3)) / P
= 2√√3 / √P
Now we know that an equilateral triangle has circumradius = S/√3,where S is the side of the triangle.
So
S/√3 = 2√√3 / √P ----------(1)
There is a unique relation between height and side of an equilateral triangle
Height(H) = (√3/2)S ----------(2)

We can solve the two equations 1 and 2 for the value of H.
But the final expression which I got after solving this is

H^2 = 9√3/P and if you solve it for H,it gives a value(>2) different from 2.
Can someone explain where I am messing up.

[vjsharma25]

OK,I think i got my answer. The question statement just states that equilateral triangle is drawn inside it,it may or may not be touching the circumference of the circle.
Hmmm,thats why we need to calculate the area of the triangle from the probability statement.

[jnelson0612]

Correct, vjsharma. Good work in figuring out your error.

[rustom.hakimiyan]

My question is regarding the Official Explanation.

If we use the 30.60.90 relationship of s:s root 3: 2s, shouldn't s be 1/2s b/c it's half of an equilatertal, which skews the whole area equation of the triangle because it makes the .5*b*h = .5(s/2)((s/2)*rt3)?

Can someone please explain to me by why the base is not s/2 and instead just s?

Thanks!

[RonPurewal]

Can someone please explain to me by why the base is not s/2 and instead just s?

Thanks!

Because the triangle for which you're ultimately calculating the area is the equilateral triangle.

You're thinking about the 30º-60º-90º triangle instead.