# Thread: Cylinder in a Sphere

1. ## Cylinder in a Sphere

Inscribed a right circular cylinder of height h and radius r in a sphere of fixed radius R. Express the volume V of the cylinder as a function of h.

HINTS GIVEN:

(1) V = pi(r^2)(h).

(2) R = the hypotenuse and r = one of the legs of the right triangle in the cylinder.

2. Originally Posted by symmetry
Inscribed a right circular cylinder of height h and radius r in a sphere of fixed radius R. Express the volume V of the cylinder as a function of h....
Hello,

I've attached a diagram (as usual )

You see that

$\displaystyle \left( \frac{1}{2}h\right)^2+r^2=R^2 \Longleftrightarrow r^2=R^2-\frac{1}{4}h^2$

Plug in this term for r&#178; in the formula for the volume:

$\displaystyle V=\pi \cdot \left( R^2-\frac{1}{4}h^2 \right) \cdot h=\pi R h-\frac{1}{4} \cdot \pi \cdot h^3$

EB

3. ## ok

How do you do that?

Thanks!

4. Originally Posted by symmetry
How do you do that?
Thanks!
Thanks for the ovation (please sit down if you are still standing).

Have a look here:
euklid dynageo homepage

EB

5. ## ok

Thank for your help and for sharing your homepage link. I will look into it this weekend.

6. ## Small Correction

In the formula ...

The first term should be (pi)(R^2)h

Otherwise, I enjoyed the wonderful illustration and solution ... very eloquent!

stevez2436

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### a right circular cylinder of height h and radius r is incribed in a right circular cone with a height of 10 ft and a base with a radius of 6 ft

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