Maximisation Problem - Cone/Sphere

**English Major** · Sep 6th 2011, 12:34 PM

Problem:
What is the volume of the largest sphere that can be placed inside a cone with a slant height of 10?

What I've done so far:

Recognize that finding the largest radius will suufice; once I find it, I can find the volume easily.
Reduce the problem to a right triangular cross-section with an inscribed semicircle. the semicircle has a diameter that lies on the height of the triangle. A radius has been drawn to the right angle vertex of the triangle, and to the hypotenuse of ten (at a right angle).

If I call the far angle across from the semicircle $\theta$ , then:

$10 sin(\theta) = r + \frac{r}{cos(\theta)}$

Or, $5 sin(2\theta) = rsin(\theta) + r$

Past this, I am stumped. An Implicit derivative doesn't seem to get me anywhere, but it is impossible to fully separate my two variables. Any thoughts?

**skeeter** · Sep 6th 2011, 03:22 PM

let $\theta$ be the base angle of the cone.

Note that a line from the center of the circle to the vertex of $\theta$ bisects $\theta$ .

$R = r \tan\left(\frac{\theta}{2}\right)$

$r = 10\cos{\theta}$

$R = 10\cos{\theta} \cdot \tan\left(\frac{\theta}{2}\right)$

find $\frac{dR}{d\theta}$ and maximize ...

I get the value $\theta \approx 52^\circ$

Thread: Maximisation Problem - Cone/Sphere

LinkBack

Thread Tools

Display

Maximisation Problem - Cone/Sphere

Re: Maximisation Problem - Cone/Sphere

Similar Math Help Forum Discussions

sphere cone problem

sphere cone problem

Sphere inscribed in cone problem

Sphere inside a cone problem.

Sphere into cone

Search Tags