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(C) Let the radius of the circular sector be $r$ and the arc length be $\ell$.Given that the total length of the wire is $20 \ m$, the perimeter of th

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$5$

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(C) Let the radius of the circular sector be $r$ and the arc length be $\ell$.Given that the total length of the wire is $20 \ m$, the perimeter of the sector is $2r + \ell = 20$.Thus, $\ell = 20 - 2r$.The area $A$ of a circular sector is given by $A = \frac{1}{2} r \ell$.Substituting the value of $\ell$, we get $A = \frac{1}{2} r (20 - 2r) = 10r - r^2$.To find the maximum area, we differentiate $A$ with respect to $r$ and set it to zero:$\frac{dA}{dr} = 10 - 2r = 0$.Solving for $r$, we get $2r = 10$, which implies $r = 5 \ m$.To verify that this is a maximum, we check the second derivative: $\frac{d^2A}{dr^2} = -2$, which is less than $0$, confirming that the area is maximum at $r = 5 \ m$.

Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed is to be the greatest (in $m$)?

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