SOLUTION: Please help me to solve this: The largest area of an equilateral triangle which can be formed by a given length of a wire is 484√3sq.cm.If the same wire is bent twice,first t

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Question 986297: Please help me to solve this: The largest area of an equilateral triangle which can be formed by a given length of a wire is 484√3sq.cm.If the same wire is bent twice,first to form a circle and then a square; find the ratio between there area
Answer: 14:11
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
TRIANGLE
Find the perimeter, same as the original length of wire.
a, altitude,
x, length of one of the equal sides
Draw the picture.
a%5E2%2B%28x%2F2%29%5E2=x%5E2
algebra steps
a=%28sqrt%283%29%2F2%29x
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Use the given area
Note that base will be x, height will be a.
We should use the entire triangle
%281%2F2%29x%2Aa=484sqrt%283%29
%281%2F2%29x%2A%28sqrt%283%29%2F2%29x=484sqrt%283%29
%281%2F4%29x%5E2=484
x%5E2=4%2A484
x=sqrt%284%2A4%2A121%29
highlight%28x=44%29
-
This means that the perimeter of the equilateral triangle is highlight%28132%2Acentimeters%29.

SQUARE
Need side length to find its area. Use known perimeter same as of the triangle.
s%2A4=132
s=132%2F4
s=33
AREA of the square, 33%5E2=highlight%281089%2Acm%5E2%29.

CIRCLE
Need radius to find its area. Use known perimeter same as of the triangle.
2pi%2Ar=132
r=132%2F%282pi%29
r=66%2Fpi
AREA of the circle is pi%2Ar%5E2=pi%2A%2866%2Fpi%29%5E2=66%5E2%2Fpi
highlight%28%284356%2Fpi%29cm%5E2%29
If you want a computed approximation for this you might start with 1386.6.

You might find a ratio of area for Square to Circle any fair way you want.
1089:1386.6 ?Square to Circle
1386.6:1089 ?Circle to Square

If you were to divide BOTH of those numbers composing the ratio by 99, you will get something extremely near to your expected 14:11. The circle-to-square ratio.
Other reportable ratios may also work well.