## 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.

Good luck! My own answer will appear in the comments. (Please show your work if you decide to submit your own answer!)

_

#### 4 comments:

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.