A sphere encloses a cube with a volume of 1000 so that all vertices of the cube touch the surface of the sphere. What is the volume of the sphere ?

Guest Dec 30, 2015

#3**+15 **

My first math answer for this year:)

Take the cuberoot of the cube volume to get the side length.

\(\sqrt[3]{1000} = 10\)

Diagonal of a cube is

\(s*\sqrt{3} = d\\ 10*\sqrt{3} = 17.32 \)

That is equal to the diameter of the sphere

Now take half that for the radius

\(\dfrac{17.32}{2} = 8.66\)

Then the volume of the the sphere is

\(\dfrac{4}{3}\pi * 8.66^3 = 2720.5 ~ units \)

So the smallest sphere that will hold a cube of 1000 is 2720.5 units

Dragonlance
Jan 1, 2016

#2**0 **

Actually this is a really good 3 dimensional question !

Some of our members may like to have a go at it!

So long as you know how to find the volume of a sphere V=(4/3) pi*r^3

and the volume of a cube then you can try it !!

HINT: You may need to use the Pythagorean Theorem too!

Melody
Jan 1, 2016

#3**+15 **

Best Answer

My first math answer for this year:)

Take the cuberoot of the cube volume to get the side length.

\(\sqrt[3]{1000} = 10\)

Diagonal of a cube is

\(s*\sqrt{3} = d\\ 10*\sqrt{3} = 17.32 \)

That is equal to the diameter of the sphere

Now take half that for the radius

\(\dfrac{17.32}{2} = 8.66\)

Then the volume of the the sphere is

\(\dfrac{4}{3}\pi * 8.66^3 = 2720.5 ~ units \)

So the smallest sphere that will hold a cube of 1000 is 2720.5 units

Dragonlance
Jan 1, 2016

#4**+5 **

Excellent answer Dragonlance but don't forget your units.

I know there are none in the question but you should add units and units^{3 } anyway

Melody
Jan 1, 2016