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

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) (Show Source): You can put this solution on YOUR website!
The proportion of the sector to a full circle determines the area:

The value of is the ratio of the arc length to the circumference of a full circle. Noting that the arc length is :

Substituting the latter into the former:

Taking the derivative of A with respect to r:

Setting this to zero:
�> (<<<�� that's an "ell" on top)

��
Check:
�> the function is concave down, thus the point dA/dr=0 we found is a maximum.

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