Problem Solving

A square and a regular hexagon have same area A . Find the ratio of the perimeter of the square to the perimeter of the hexagon.
A) sqrt3 :2
B) 1 : 2
C) 3^(1/4) : 4^(1/4)
D) 4^(1/4) : 3^(1/4)
E) None

to me it seems answer is E

assuming square side = a, Hexagon size = b

a^2 = (3 * sqrt 3)/2 * b^2

=> 4a/6b = 2/3 * sqrt [ (3* sqrt 3) /2)]
or can be put this way: 4a/6b = (sqrt 2)/ 3 * [sqrt ( 3 * sqrt 3)]


its not matching with option A, B, C..

OR, am i doing something wrong?

any other thoughts??

OA is D ..
Area of regular hexagon of side a= 6x sqrt3 /4 x a^2
Area of square of side b = b^2
6x sqrt3 /4 x a^2 = b^2
b/a = [3)^(3/4) ]/sqrt 2
ratio of perimeter of square to hexagon = 4b/6a = 4/6 x[[3)^(3/4) ]/sqrt 2] =[spoiler] (4/3)^(1/4) Ans D[/spoiler]

gmattarget700 wrote:to me it seems answer is E

assuming square side = a, Hexagon size = b

a^2 = (3 * sqrt 3)/2 * b^2

=> 4a/6b = 2/3 * sqrt [ (3* sqrt 3) /2)]
or can be put this way: 4a/6b = (sqrt 2)/ 3 * [sqrt ( 3 * sqrt 3)]


its not matching with option A, B, C..

OR, am i doing something wrong?

Hey gmattarget700,
This is what you have written:-

4a/6b = 2/3 * sqrt [ (3* sqrt 3) /2)]

Now look carefully at the answer choices:-They are only given to confuse you.
I will explain step-by-step
2/3 * sqrt [ (3* sqrt 3) /2) can be written as (2/3)[(3^1/2)(3^1/4)/(2^1/2)]
Cancelling the exponents of 2 and 3 in Dr. and Nr.respectively ,
=>(2^1/2)/(3^1/4)
Now 2=4^1/2
Hence we get [(4^1/2)^1/2}/(3^1/4)
which is equal to (4^1/4)/(3^1/4)
which is C.

I hope the explanation helps.

I realized that after seeing the OA.... looking at None choice made me jump to decide on it faster.....yeah, need to be more careful...as it clearly shows that even if you have done right calculation and have right answer, you may not pick the right choice

thanks anyway, for your better explanation...

yeah u did some mistake in between and thats why u went far away from ans...

I am interested to know which " some mistake in between" i did, as you mentioned....I had deduced the RIGHT answer (4a/6b = 2/3 * sqrt [ (3* sqrt 3) /2)]...Harsh also quoted that) but it needed some "massaging" to map with right choice but I didn't have patience to do that massaging and so wrongly chose E.

so tell me which mistake you are referring??

gmattarget700 wrote:I am interested to know which " some mistake in between" i did, as you mentioned....I had deduced the RIGHT answer (4a/6b = 2/3 * sqrt [ (3* sqrt 3) /2)]...Harsh also quoted that) but it needed some "massaging" to map with right choice but I didn't have patience to do that massaging and so wrongly chose E.

so tell me which mistake you are referring??

I was saying about the same gmattarget 700..... well u have corrected it now...