SOLUTION: A piece of string l is bent to form the sector of a circle radius r. Show that the area of the sector is maximised when r=1/4 l

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Question 1128389: A piece of string l is bent to form the sector of a circle radius r. Show that the area of the sector is maximised when r=1/4 l
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
The proportion of the sector to a full circle determines the area:
+A+=+%28theta%2F360%29%28pi%29r%5E2+

The value of +theta+ is the ratio of the arc length to the circumference of a full circle. Noting that the arc length is +l-2r+:
+theta+%2F+360+=++%28l-2r%29+%2F+%282%28pi%29r%29+

Substituting the latter into the former:
+A+=+%28%28l-2r%29%2F2%28pi%29r%29+%28%28+pi%29+r%5E2+%29+
+A+=+%28lr%2F2%29+-+r%5E2+

Taking the derivative of A with respect to r:
+dA%2Fdr+=+l%2F2+-+2r+

Setting this to zero:
+l%2F2+-+2r+=+0+ —> +highlight%28+r+=+l%2F4+%29+ (<<<—— that's an "ell" on top)

——
Check:
+d%5E2A%2Fdr%5E2+=+-2+ —> the function is concave down, thus the point dA/dr=0 we found is a maximum.