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1. Two identical solid spheres fit exactly inside a cuboid box. Find the percentage of the volume of the box which is occupied by the spheres. Give your answer to 1dp.
I've already figured out the formula for surface area of a sphere (4/3pie r^2) and I know the height would be 4r and the base 2r (diameter), although I'm unsure of what to do from here.
ANY HELP WOULD BE APPRECIATED
2. (Original post by neharajput)
Two identical solid spheres fit exactly inside a cuboid box. Find the percentage of the volume of the box which is occupied by the spheres. Give your answer to 1dp.
I've already figured out the formula for surface area of a sphere (4/3pie r^2) and I know the height would be 4r and the base 2r (diameter), although I'm unsure of what to do from here.
ANY HELP WOULD BE APPRECIATED
Aren't you supposed to work out the volume of each sphere rather than the surface area?
(And another thing I'd do would be to somehow relate measurements on the spheres to the width, length and height of the cuboid.)
3. (Original post by TheOtherSide.)
Aren't you supposed to work out the volume of each sphere rather than the surface area?
(And another thing I'd do would be to somehow relate measurements on the spheres to the width, length and height of the cuboid.)
I meant volume rather than surface area, sorry but from there I'm not sure what to do, as I have related the lengths of the cuboid in terms of the radius of the sphere. Could you guide me as to what to do after this?
The height would be 4 x radius (4r) and the base 2r (2 x radius)
4. (Original post by neharajput)
I meant volume rather than surface area, sorry but from there I'm not sure what to do, as I have related the lengths of the cuboid in terms of the radius of the sphere. Could you guide me as to what to do after this?
The height would be 4 x radius (4r) and the base 2r (2 x radius)
Don't forget that since the sphere is a 3D shape, the width would also be 2r. Using these measurements you have for the cuboid, you should then have a value for the volume in terms of r.
5. (Original post by TheOtherSide.)
Don't forget that since the sphere is a 3D shape, the width would also be 2r. Using these measurements you have for the cuboid, you should then have a value for the volume in terms of r.
So the volume of the cuboid would be 16r^3 because 2r x 4r x 2r
Then the volume of the two spheres would be 8/3 x pie^2 x r^4
I don't understand what's next, as r is an unknown value...
6. (Original post by neharajput)
So the volume of the cuboid would be 16r^3 because 2r x 4r x 2r
Then the volume of the two spheres would be 8/3 x pie^2 x r^4
I don't understand what's next, as r is an unknown value...
Well, since finding the percentage is the same as dividing one value by another and then multiplying by 100, you could make a fraction from the volume for the two spheres and the volume of the cuboid.
Spoiler:
Show
By the way, after reading the question again, I was wondering if you were sure that there weren't any other values stated about either the radius of the sphere or something similar?
7. (Original post by TheOtherSide.)
Well, since finding the percentage is the same as dividing one value by another and then multiplying by 100, you could make a fraction from the volume for the two spheres and the volume of the cuboid.
Spoiler:
Show
By the way, after reading the question again, I was wondering if you were sure that there weren't any other values stated about either the radius of the sphere or something similar?
No there wasn't any values stated, just that the radius is r...
8. If you think about it, however much you scale the box & spheres, the ratio of sphere volume to box volume is never going to change. Double it, you double both volumes, so the percentage is the same.

This means it's going to cancel out. The volume of the spheres will be some multiple of r cubed, and the volume of the spheres should be some multiple of r cubed. r cubed/r cubed = 1, so it's just the ratio of the multiples.

It's also exactly the same as one sphere in a box, as the two "sides" of the box are identical, so have the same ratio.
9. (Original post by Hanvyj)
If you think about it, however much you scale the box & spheres, the ratio of sphere volume to box volume is never going to change. Double it, you double both volumes, so the percentage is the same.

This means it's going to cancel out. The volume of the spheres will be some multiple of r cubed, and the volume of the spheres should be some multiple of r cubed. r cubed/r cubed = 1, so it's just the ratio of the multiples.

It's also exactly the same as one sphere in a box, as the two "sides" of the box are identical, so have the same ratio.
But wouldn't the volume of the spheres together be r^4 because r^2 x r^2
10. (Original post by neharajput)
But wouldn't the volume of the spheres together be r^4 because r^2 x r^2
Actually, it wouldn't be - you're adding the two volumes together to get the total volume of the spheres.
11. (Original post by TheOtherSide.)
Actually, it wouldn't be - you're adding the two volumes together to get the total volume of the spheres.
Ohh, I just realised I have been writing the formula wrong the whole time:')
So would the equation be...
16r^3/ (8/3 x 2pie x 2r^2)?
12. Spoiler:
Show
it would be (8/3)(pi)(r^3) as you are multiplying (4/3)(pi)(r^3) by 2. therefore, to find the percentage of the volume of the cuboid that the spheres occupy you take the volume of the spheres, (8/3)(pi)(r^3), divide by the volume of the cuboid, 16r^3, then multiply by 100. the r^3 should cancel out leaving you with a number that should be something long the lines of 12800pi/3 <-- my mental arithmetic may be wrong with that last bit though as I don't have a calculator to hand EDIT:my mental arithmetic was definitely wrong, i meant 800pi/48
13. 1)volume of a sphere is 4/3 pi r^3, not r^2
2)you want the volume of the spheres as a percentage of the volume of the cuboid so you should be using the reciprocal of what you just wrote once you have corrected the sphere volume part of the formula.
14. (Original post by neharajput)
Ohh, I just realised I have been writing the formula wrong the whole time:' )
So would the equation be...
16r^3/ (8/3 x 2pie x 2r^2)?
Here, I think you've multiplied the whole formula by 2 too many times - it should just be 8/3 pi r^3
15. (Original post by spico)
Spoiler:
Show
it would be (8/3)(pi)(r^3) as you are multiplying (4/3)(pi)(r^3) by 2. therefore, to find the percentage of the volume of the cuboid that the spheres occupy you take the volume of the spheres, (8/3)(pi)(r^3), divide by the volume of the cuboid, 16r^3, then multiply by 100. the r^3 should cancel out leaving you with a number that should be something long the lines of 12800pi/3 <-- my mental arithmetic may be wrong with that last bit though as I don't have a calculator to hand
I worked it out and got 52.36%???
16. checked on a calculator, you're right with that answer, i screwed the arithmetic up. question wants 1dp though
17. (Original post by spico)
checked on a calculator, you're right with that answer, i screwed the arithmetic up. question wants 1dp though
yayy) THANKYOU SOOO MUCH really helped
18. (Original post by TheOtherSide.)
Actually, it wouldn't be - you're adding the two volumes together to get the total volume of the spheres.
I found the answer, but thankyouuuu soon muchhhh)

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