## Tuesday, May 15, 2012

### this week's MGRE Math Beast Challenge

From here:

The perimeter of an equilateral triangle is 1.25 times the circumference of a circle.

Quantity A
The area of the equilateral triangle

Quantity B
The area of the circle

(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.

By the way, sorry about the lack of posting last week. I think my sickness lingered on a bit longer than I had thought.

_

#### 1 comment:

Kevin Kim said...

I'm going to give my vote to (B): the area of the circle is greater.

Why?

I worked it out this way:

Assume an equilateral triangle with side x.

Assume a circle with radius r.

So:

3x = 1.25(2πr)

3x = 2.5πr

r = 3x/(2.5π)

The area of the circle, then, is:

π•((3x/(2.5π))^2) =

(9x^2)/(6.25π)

The area of the equilateral triangle is:

((x^2)√3)/4

Let "Q" mean "greater than, less than, equal to (as yet undetermined)."

We need to compare areas:

[A(circle)] Q [A(triangle)]

[(9x^2)/(6.25π)] Q [((x^2)√3)/4]

Divide both sides of the Q-relation by x^2:

9/(6.25π) Q (√3)/4

Once we evaluate these expressions, we see the decimal approximations are:

.4583662 > .4330127

The area of the circle, then, is larger by a small margin.

QED.

By the way, this one would be hard to solve without a calculator, unless you happen to have memorized the decimal approximation for √3.