Question

Set up an equation of a tangent to the graph of the following function.
A sector with a central angle $\alpha$ is cut off from a circle. A cone is made of the remaining part of the circle. At what value of $\alpha$ is the capacity of the cone the greatest?

Answer 1

Consider the cone

We know $S$ and $T$, Then

$r=S\ast T/(2\ast pi)⟶\left(1\right)$

$h=\sqrt{S^2-r^2}⟶\left(2\right)$

$t=Arc\tan \left(\frac{r}{h}\right)⟶\left(3\right)$

So $S=24$ inches

We are solving for the central angle $T=\theta$ which will maximum the volume $V=\frac{{\pi r}^{2}h}{3}$

Therefore, $V=\frac{\pi \frac{S^2{T}^2}{{4\pi }^2}\times \sqrt{S^2-{\left(\frac{ST}{2\pi }\right)}^2}}{3}$

$\frac{dV}{dt}=0$

$\frac{2\theta S^3}{4\pi }\sqrt{\frac{(4{\pi }^2-{\theta }^2)}{{4\pi }^2}}=\frac{{\theta }^3{S}^3}{4\pi \sqrt{\frac{4{\pi }^2{\theta }^2}{{4\pi }^2}}}$

${\theta }^2=\frac{{4\pi }^2}{{4\pi }^2+1}$

$\theta =\sqrt{\frac{4{\pi }^2}{{4\pi }^2+1}}$

$\theta =0.9875$ radians

$\theta ={56.58}^0$.