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 captainglyde February 19th, 2008 08:46 AM

area of eq. triangle vs. area of square

Without a calculator, using simple mental arithmetic and approximations, I should be able to solve the following problem but I'm having a lot of trouble:

One of two ropes equal in length is cut into three segments to form the largest possible triangular area. The other rope is cut into four segments to form the largest possible rectangular area. Which of the following most closely approximates the ratio of the triangle's area to the rectangle's area?

1:2
2:3
3:4
1:1
4:3

(ans: 3:4)

Here's how I try to solve this problem but get the wrong answer. If anyone could show me what I'm doing wrong, I'd appreciate it... thank you...

if rope is length x, then we have an equilateral triangle of side length (x/3) and a square of side length (x/4).

Thus, respective areas should be (x/3)^2sqrt3/4 and (x/4)^2.

Simplifying gives x^2sqrt3/9x4 and x^2/16.

Multiply both by 4/x^2 should leave sqrt3/9 versus 1/4.

How do I look at those numbers and with simple arithmetic say "sqrt3/9:1/4 is approximately 3:4?"

Thank you very much!

 CRGreathouse February 19th, 2008 08:55 AM

Re: area of eq. triangle vs. area of square

Quote:
 Originally Posted by captainglyde How do I look at those numbers and with simple arithmetic say "sqrt3/9:1/4 is approximately 3:4?"
Well, sqrt(3) is between 3/2 and 2 since 9/4 < 3 < 4. The ratio is thus between 1.5/9:1/4 = 1/6:1/4 = 4:6 = 2:3 and 4/9:1/4 = 16:9. These are both close to 3:4.

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