# how to find maximum volume of cube inside cone

• Jun 13th 2009, 05:24 PM
Gul
how to find maximum volume of cube inside cone
I am finding this one tricky.

Find the maximum volume of cube inscribed a right cone with radius 1, and height 3.

Why can't I do 3/1 = 3-x/x

If this was a cylinder, I'd have 3/1 = 3-h/r, and with a cube the sides are equal.
• Jun 13th 2009, 06:06 PM
mr fantastic
Quote:

Originally Posted by Gul
I am finding this one tricky.

Find the maximum volume of cube inscribed a right cone with radius 1, and height 3.

Why can't I do 3/1 = 3-x/x

If this was a cylinder, I'd have 3/1 = 3-h/r, and with a cube the sides are equal.

Assuming you have defined x to be the sidelength of the cube then $\displaystyle \frac{3}{1} = \frac{3 - x}{\frac{x}{2}} = \frac{2(3 - x)}{x}$.
• Jun 13th 2009, 06:11 PM
Gul
why is it a 1/2 - x/2, and not just x/2?
• Jun 13th 2009, 06:15 PM
mr fantastic
Quote:

Originally Posted by Gul
why is it a 1/2 - x/2, and not just x/2?

You're right. I mixed up my triangles and made a typo (too busy being funny to be careful). It's fixed.
• Jun 13th 2009, 06:21 PM
Gul
They get this:

3-X / 3 = Xsqrt(2) /2
• Jun 13th 2009, 07:15 PM
mr fantastic
Quote:

Originally Posted by Gul
They get this:

3-X / 3 = Xsqrt(2) /2

Oboy. It's my day for dumb mistakes today. I have no tme now but I'll show where that comes from later.

Edit: The key is to note that when the cube is fit inside a cone, the longest side
is the diagonal of its top face. Read:

http://www.math.hawaii.edu/home/putnam/1998.pdf
• Jun 13th 2009, 07:44 PM
Gul
Thanks!, just one question, why can you do the ratio I had before for a cylinder, but not for a cube
• Jun 13th 2009, 10:54 PM
mr fantastic
Quote:

Originally Posted by Gul
Thanks!, just one question, why can you do the ratio I had before for a cylinder, but not for a cube

The cross-section of a cylinder is a circle. The cross-section of a cube is a square. The distance from the centre of a circle to any point on the perimeter is always the same. This is not the case for a square.
• Jun 13th 2009, 11:07 PM
Gul
Thank you. Are there any other rules which are important for inscribed solids?