A wire is cut into two pieces, one of length a and the other of length b. The piece of length a is bent to form an equilateral triangle, and the piece of length b is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is a/b?
+128407 The side of the equilateral triangle = a/3
So....the area of the equilateral triangle = (1/2)(a/3)^2 sqrt (3) / 2 = a^2 [ sqrt (3)/ 36 ]
The side of the hexagon = b / 6
So...it's area = 6 (1/2) (b/6)^2 sqrt (3) / 2 = b^2 [ 3 sqrt (3) / 72] = b^2 [sqrt (3) / 24]
So
a^2 [ sqrt (3) / 36 ] = b^2 [ sqrt (3) / 24]
So
a^2 / b^2 = [ sqrt (3) /24 ] / [ sqrt (3) / 36 ]
a^2 / b^2 = [ 36 / 24 ]
a^2/ b^2 = 3/2
a/ b = sqrt (3) / sqrt (2) = sqrt (6) / 2
