## Monday, December 26, 2011

### this week's MGRE Math Beast Challenge problem

From here:

This Week's Problem: "Honeycomb"

The honeycomb figure above consists of six identical regular hexagons, each with side length 2√3.

Quantity A
The total area of the honeycomb

Quantity B
180

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

_

Charles said...

Unless I'm missing something (which is entirely within the realm of possibility), the answer is B: Quantity B is greater.

Each regular hexagon is composed of six equilateral triangles with side length 2√3. The area of an equilateral triangle is (side length squared * √3) / 4, which means that the area of each equilateral triangle is 3. 6 triangles makes 18, 6 hexagons makes 108.

Kevin Kim said...

You probably caught this already, but the area of each equilateral triangle is actually going to be 3√3.

Kevin Kim said...

Sorry about the brief comment; I was at work when I wrote it, and I hate typing lengthy messages on my Droid.

To elaborate:

Given that the area of an equilateral triangle is

(s^2(√3))/4

and that, in this case,

s = 2√3

then

s^2 = (2√3)^2 = 12

which means that

(s^2)√3 = 12√3

and

((s^2)√3)/4 = (12√3)/4 = 3√3.

That being the case, each hexagon's area is (3√3)*6 =

18√3.

√3 is approx. 1.732, so 18√3 is approx.:

31.18, or 31.2.

Quantity B is 180; since there are six equal hexagons, each hexagon's area is exactly 30.

By my reckoning, then, Quantity A is greater than Quantity B, so the answer is (A).

(The new GRE allows you to use an on-screen calculator, which would be useful in this case.)

Charles said...

Whoops. Dropped a sqrt 3. Too many square roots flying around. Should've used pen and paper to keep track.