# cylinder n sphere

• Aug 10th 2010, 06:15 AM
furor celtica
cylinder n sphere
a circular cylinder is to fit inside a sphere of radius 10cm. calculate the maximum possible volume of the cylinder. (it is probably best to take as your indpendent variable the height or half the height of the cylinder).
i'm stuck here cos i don't know where to start. i drew a section of the both to try and find a clue but i'm lost. any hints?
• Aug 10th 2010, 07:12 AM
Soroban
Hello, furor celtica!

Quote:

A circular cylinder is inscribed in a sphere of radius 10 cm.
Calculate the maximum possible volume of the cylinder.

Code:

              * * *           *          *         * - - - + - - - *       *|      |      |*         |      |      |       * |      |      | *       * |      *      | *       * |      | * 10  | *         |      y|  *  |       *|      |  x  * |*         * - - - + - - - *           *          *               * * *

The radius of the cylinder is $\displaystyle x.$
The height of the cylinder is $\displaystyle 2y.$

We see that: .$\displaystyle x^2+y^2\:=\:10^2 \quad\Rightarrow\quad y \:=\:\sqrt{100-x^2}$

The volume of the cylinder is: .$\displaystyle V \;=\;\pi r^2h$

So we have: .$\displaystyle V \;=\;\pi x^2(2y)$

. . . . . . . . . .$\displaystyle V \;=\;2\pi x^2(100-x^2)^{\frac{1}{2}}$

And that is the function we must maximize.

• Aug 10th 2010, 07:41 AM
AllanCuz
Quote:

Originally Posted by Soroban
Hello, furor celtica!

Code:

              * * *           *          *         * - - - + - - - *       *|      |      |*         |      |      |       * |      |      | *       * |      *      | *       * |      | * 10  | *         |      y|  *  |       *|      |  x  * |*         * - - - + - - - *           *          *               * * *
The radius of the cylinder is $\displaystyle x.$
The height of the cylinder is $\displaystyle 2y.$

We see that: .$\displaystyle x^2+y^2\:=\:10^2 \quad\Rightarrow\quad y \:=\:\sqrt{100-x^2}$

The volume of the cylinder is: .$\displaystyle V \;=\;\pi r^2h$

So we have: .$\displaystyle V \;=\;\pi x^2(2y)$

. . . . . . . . . .$\displaystyle V \;=\;2\pi x^2(100-x^2)^{\frac{1}{2}}$

And that is the function we must maximize.

You know Soroban...I saw this topic and got all giddy when there were no responses. Now that I am no longer the first, I no longer feel so giddy! Thanks for taking that away (Evilgrin) lol

I would solve it a little bit differently.

Equation of the sphere $\displaystyle z^2 + x^2 + y^2 = 10^2$

Equation of the cylinder $\displaystyle x^2 + y^2 = a^2$

Subbing the cylinder equation into the sphere equation,

$\displaystyle z^2 + a^2 = 100 \to z= \sqrt{ 100 -a^2 }$

Transforming to cylindrical co-ordinates,

$\displaystyle V = 2 \int_0^{ 2 \pi } d \theta \int_0^a rdr \int_0^{ \sqrt{100-a^2} } dz$

Compute the above and find $\displaystyle \frac{dV}{dA} = 0$

We would find this to be the same equation as that of Soroban, but I always like to derive max volume questions via the triple integral representation. It's good practice I feel.