Minimum volume word problem

Mario1776

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Aug 11, 2006
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Sorry for this being vague but it is all the info I have.

You have a sphere with radius r. The sphere has to fit inside of a cone. What would be an equation to figure out the dimensions of the cone (with the cone having the least volume possible) relative to the sphere being inside it, an equation to figure out the minimum volume of the cone, and the equation to figure out the minimum volume of the cone minus the volume of the sphere.

Any help that could be given would be greatly appreciated.
 
"inside"

The sphere has to fit inside the cone, such that the bottom of the sphere is even with the bottom of the cone.

Here is my very poor two dimensional artistic rendering:
conespherede4.png
 
Did you look at the link provided in an earlier reply?

Eliz.
 
This is a classic max/min problem.

The variations may be cone in a sphere, sphere in a cone, sphere in a cylinder, cylinder in a sphere. These come up often in calc classes.

Nevermind the link, here's how I would do it:

You can use similar triangles.

See triangle AOB and big triangle ADC?.

Use those.

Let h=AD, r=OB=OD, R=DC, h-r=AO

\(\displaystyle \L\\\frac{R}{h}=\frac{r}{\sqrt{(h-r)^{2}-r^{2}}}\)

Now, solve this for r, sub into the volume formula for a cone, differentiate with respect to h, set to 0 and solve for h

That's what I done and got \(\displaystyle \L\\h=\sqrt{3}R\;\ and \;\ r=\frac{R}{\sqrt{3}}\)

You can use these results to find the ratio between the sphere's volume and the cone's volume. It'll be something nice.
 
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