A cylindrical tank has a base with a circumference of 4(√(�√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
Circumference of the base:
2�r = 4√(�√3)
r = 4√(�√3) / 2�
= 2√(�√3) / �.
Area of the base:
�r² = �* [2√(�√3) / �]²
= � * 4�√3 / �²
= 4√3.
Area of the triangle:
Since P(outside triangle) = 3/4, P(triangle) = 1/4.
Thus, the area of the triangle = 1/4(circle area) = (1/4) * 4√3 = √3.
The formula for the area of an equilateral triangle = (s²√3)/4.
Thus:
(s²√3)/4 = √3
s² = 4
s=2.
The correct answer is
E.
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