SOLUTION: A cube and a sphere would have the same volume if the sphere’s radius were halved. What is the exact ratio of the sphere’s radius to the cube’s side?

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Question 257194: A cube and a sphere would have the same volume if the sphere’s radius were
halved. What is the exact ratio of the sphere’s radius to the cube’s side?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
if the cube and the sphere have the same volume, then the following equation is true.

s%5E3+=+%284%2F3%29%2Api%2Ar%5E3

solve for s to get:

s+=+root%283%2C%28%284%2F3%29%2Api%2Ar%5E3%29%29

this becomes:

s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Aroot%283%2C%28r%5E3%29%29%29

which becomes:

s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar

to confirm this is correct, we supply any value for r and calculate the volume of the sphere and the volume of the cube.

let r = 6.

volume of the sphere is equal to %284%2F3%29%2Api%2A6%5E3 = 904.7786842 cubic units.

side of the cube = root%283%2C%28%284%2Api%29%2F3%29%29%2A6 = 9.671951724.

volume of the cube = (9.671951724)^3 = 904.7786842.

volumes are the same so the formula is good.

question was:

What is the exact ratio of the sphere’s radius to the cube’s side?

the original statement was?

A cube and a sphere would have the same volume if the sphere’s radius were
halved.

since these cubes have the same volume, this means that the radius used was half the radius of the sphere.

this means that the sphere's radius is double whatever value we used to make the volumes equal.

in our formula for s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar, we'll replace "r" with "r/2" to get:

s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%28r%2F2%29


this means that in our example, the radius of the sphere was actually 12 rather than 6.

plug that into the revised formula and we get:

s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%2812%2F2%29 which becomes:

s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%286%29 which is the same we had before when the volumes were equal.

so our formula is:

s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%28r%2F2%29 and we want to get the ratio of r to s.

in order to find that ratio, we want to get r/s = ???

we multiply both sides of our formula by 2 to get:

2%2As+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar

we divide both sides of our formula by root%283%2C%28%284%2Api%29%2F3%29%29 to get:

2%2As%2Froot%283%2C%28%284%2Api%29%2F3%29%29+=+r

we divide both sides of this formula by s to get:

2%2Froot%283%2C%28%284%2Api%29%2F3%29%29+=+r%2Fs

our ratio is:

r%2Fs+=+2%2Froot%283%2C%28%284%2Api%29%2F3%29%29

we confirm this is true by taking any value for s and seeing if the formula holds.

we multiply both sides of this ratio by s to get:

r+=+%282%2As%29%2Froot%283%2C%28%284%2Api%29%2F3%29%29

when we let s = 5, this formula becomes:

r+=+%2810%29%2Froot%283%2C%28%284%2Api%29%2F3%29%29 = 6.203504909

we take half of this to get half of r = 3.101752454

volume of the cube is 5%5E3+=+125

volume of the sphere when the radius is halved is %284%2F3%29%2Api%2A%283.101752454%29%5E3+=+125

we're good.

your answer is:

the ratio of the radius of the sphere to the side of the cube is:

r%2Fs+=+2%2Froot%283%2C%28%284%2Api%29%2F3%29%29

this is the ratio of the radius of the sphere before we halved it, as I believe that's what they are asking.