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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 ?

 Dec 30, 2015

Best Answer 

 #3
avatar+1312 
+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

 Jan 1, 2016
 #1
avatar
+5

i found thanks ! (250pi sqrt(2)

 Dec 30, 2015
 #2
avatar+105509 
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!

 Jan 1, 2016
 #3
avatar+1312 
+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
avatar+105509 
+5

Excellent answer Dragonlance but don't forget your units.

I know there are none in the question but you should add units and units3  anyway   laugh

 Jan 1, 2016
 #5
avatar+104712 
+5

Excellent thinking, Dragonlance.......!!!!!

 

Here's a pic of the cross-section of the cube within the sphere showing that Dragonlance's answer is indeed correct.....!!!

 

 

 

 

 

cool cool cool

 Jan 1, 2016
edited by CPhill  Jan 1, 2016
 #6
avatar+105509 
+5

Thanks Chris, 

This is a clip to show you how to find the legth of the diagonal of the cube

https://www.youtube.com/watch?v=UN8wB0KdmiQ

 Jan 2, 2016

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