If a liquid from a completely filled spherical container of inner radius #r# is poured into a cube shaped container, what would be the dimensions of the cube in terms of the radius of the sphere?

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Answer 1 · 2017-09-28T10:09:15

#root(3)(36pi)/3r#

Volume of cube is equal to volume of sphere.

Volume of a sphere: #V = 4/3pir^3#

Let #a# be a side of the cube.

Then:

#a^3# = volume of cube.

So we have:

Volume of cube = volume of sphere:

#a^3 = 4/3pir^3#

We need to manipulate #4/3pir^3# to get the required result.

Since #4/3pir^3# can be expressed #(4pir^3)/3# This is just a fraction like any other, so we can multiply numerator and denominator by #9#

This then gives:

#(36pir^3)/27#

Now we have:

#a^3 = (36pir^3)/27#

Taking cube roots:

#root(3)(a^3) = root(3)((36pir^3)/27) => root(3)(a^3) = (root(3)(36pir^3))/(root(3)(27))#

Extracting and evaluating any cubes:

#a = (rroot(3)(36pi))/3#

Which can be expressed:

#a = (root(3)(36pi))/3r#