20 meters of wire is available to fence of a flowerbed in the form of a circular sector. If the flowerbed is…
20 meters of wire is available to fence of a flowerbed in the form of a circular sector. If the flowerbed is to have maximum surface area, then the radius of the circle is
$8 \mathrm{~m}$
$4 \mathrm{~m}$
$2 \mathrm{~m}$
$5 \mathrm{~m}$
Solution
$\theta^c=\frac{\ell}{r}=\frac{20-2 r}{r}=\frac{20-2 r}{r} \times \frac{180^{\circ}}{\pi}$
Now area of sector $=\frac{\pi r^2 \theta}{360^{\circ}}$
$A(r)=\frac{\pi r^2 \times \frac{20-2 r}{r} \times \frac{180^{\circ}}{\pi}}{360^{\circ}}=10 r-r^2$
For maximum area $A^{\prime}(r)=0$
$\begin{aligned} & \Rightarrow 10-2 r=0 \\ & \Rightarrow r=5 \mathrm{~m}\end{aligned}$
For maximum area $A^{\prime}(r)=0$
$\begin{aligned} & \Rightarrow 10-2 r=0 \\ & \Rightarrow r=5 \mathrm{~m}\end{aligned}$