A cylindrical tank has a base with a circumference of 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"

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).

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

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

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.

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.