Really Cutting off the Thing

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The radius of a cone is square root 2 ( \sqrt{2} )times the height of the cone. A cube of maximum volume is cut from the same cone.What is the ratio of the volume of the cone to the volume of the cube ?

2.25 . Pie 2.35 3.18 . Pie can't be determined

2 solutions

Ujjwal Rane
Aug 21, 2016

Max Volume Cube in Cone

We can solve it using just a 2D projection instead of the 3D solids. The 'Front View' of the cone would be a triangle of height (h) taken to be unity without any loss of generality. So the base would be $2\sqrt{2}$ and its volume would be $\frac{2 \pi}{3}$

The cube will be placed on the base of the cone with its top vertices meeting the cone. Thus the circumscribing circle of the cube face will be the cross section of the cone at that elevation.

Take origin at the center of the cube. The cone slant height in the intercepts form will be the line $\frac{x}{\sqrt{2}} + \frac{y}{1} = 1$ A line $y = \sqrt{2}x$ will meet the cone at the desired elevation (y). Solving the two y = 2/3 which is the height and therefore the side of the cube! With volume = $(\frac{2}{3})^3$

So the ratio will be $\frac{9}{4} \pi = 2.25 \pi$

How is the slope=√2?

Krishnendu Mondal - 3 years, 8 months ago

Consider a cube (say unit cube for simplicity of calculations) and imagine the line going from the center of its bottom face to one of the vertices of the top face. It will go up by (rise) one unit and at the same time go horizontally (the run) 1/sqrt(2). Thus its slope Rise/Run turns out to be sqrt(2)

Ujjwal Rane - 3 years, 7 months ago

Yog Sharma
Jan 4, 2018

I don't know

Really Cutting off the Thing

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