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# Sphere

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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

#3
+15

My first math answer for this year:)

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

$$\sqrt{1000} = 10$$

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
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i found thanks ! (250pi sqrt(2)

Dec 30, 2015
#2
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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
+15

My first math answer for this year:)

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

$$\sqrt{1000} = 10$$

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

I know there are none in the question but you should add units and units3  anyway Jan 1, 2016
#5
+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.....!!!    Jan 1, 2016
edited by CPhill  Jan 1, 2016
#6
+5

Thanks Chris,

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