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Carrot182
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#1
Report Thread starter 16 years ago
#1
Im really stuck on what to do on my math guttering coursework.
Its about finding the largest area of a cross section with a 20cm length of guttering. I need to investigate different shapes like rectanges, triangles, polygons and semi-circles. I know that the semi-circle has the largest area but i need to show how and why.

any help would be appreciated

Thanx

Phillippa
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sazzles
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#2
Report 16 years ago
#2
This isn't really a proof, but it's the best I can think of at the moment.
Let the perimeter of all your shapes equal P and calculate the area in the form A=kP^2, where k is a constant depending on the shape. The shape with the highest value of k will have the largest area for a given perimeter.

eg1. Circle, radius r
P=2πr
Rearrange to get r;
r=P/2π
Substitute this into the equation for the area of a circle;
A=πr^2
A=π(P/2π)^2
A=P^2/4π^2
For a circle, k=1/4π=0.0795 (3sf)

eg2. Semi-circle, radius r
P=πr+2r
Rearrange to get r;
r=P/(π+2)
Substitute into the equation for the area of a semi-circle;
A=πr^2/2
A=π(P/(π+2))^2/2
A=(πP^2)/2(π^2+4π+4)
For a semi-circle, k=π/2(π^2+4π+4)=0.0594 (3sf)

eg3. Square, side length l
P=4l
Rearrange to get l;
l=P/4
Substitute into the equation to get the area of a square;
A=l^2
A=(P/4)^2
A=P^2/16
For a square, k=1/16=0.0625 (3sf)

etc
I don't understand why you say a semi-circle is the biggest, a square or a circle seem to be larger?

Hope this is of some use. But I advise you to check my calculations carefully-I often make mistakes.

(If my symbols are confusing π=pi and x^2=x squared)

This link should be useful;
http://mathforum.org/library/drmath/view/53384.html
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