SOLUTION: A square and an equilateral triangle both have area of 16√3.Find the ratio of the length of the square side to the length of the triangle side. (Round your answer to the near

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Question 288606: A square and an equilateral triangle both have area of 16√3.Find the ratio of the length of the square side to the length of the triangle side. (Round your answer to the nearest hundredth)
A) 0.25 B) 0.66 C) 1.52 D) 4 E) none of these
Answer by dabanfield(803) About Me  (Show Source):
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A square and an equilateral triangle both have area of 16√3.Find the ratio of the length of the square side to the length of the triangle side. (Round your answer to the nearest hundredth)
A) 0.25 B) 0.66 C) 1.52 D) 4 E) none of these
Let x be a side of the square and y be a side of the triangle. Then we have:
area of the square = x^2 = 16*sqrt(3)
1.) x = 4*3^(1/4)
area of the triangle = 1/2*base*height
Let h be the altitute of the equalateral triangle. The altitute forms two right triangles. By the Pythagorean Theorem we have then:
y^2 = (y/2)^2 + h^2
h^2 = y^2 - (y/2)^2
h^2 = (3/4)*y^2
The area of the triangle then is:
(1/2)*y*(3/4)*y^2) = 16*sqrt(3)
(3/8)*y^3 = 16*sqrt(3)
y^3 = (8/3)*16*sqrt(3)
2.) y^3 = (128/3)*sqrt(3)
Note that x/y = x^3/y^3.
From 1.) we have x^3 = (4*3^1/4)^3
x/y = x^3/y^3 = (64*(3^(3/4))/((128/3)*sqrt(3))