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## Homework Statement:

- What volume of material is removed from a solid sphere of radius 2r by drilling a hole of radius r through the center

## Relevant Equations:

- integration formula

The volume of the sphere = [itex] \frac{32{\pi}r^{3}}{3}[/itex]

The answer given at the back of the book is [itex] (\frac {32}{3} - 4\sqrt{3}){\pi}r^3[/itex]

To drill a hole completely through the sphere, the hole would have to have a length of 4r.

To get the answer in the back of the book, it requires setting the length of the hole to [itex] r = \sqrt{4r^2-x^2} [/itex] and solving for r. So the hole has a radius of r, and a length of [itex]2\sqrt3r[/itex]

But this doesn't account for the material of the sphere that is greater than [itex]x=\sqrt3r[/itex] and less than [itex]x=-\sqrt3r[/itex]

Not only that, but the answer seems to be the volume of the sphere - the volume of a hole that doesn't completely go through the sphere. The question says "what volume of material is removed," The subtraction mentioned previously give you the volume of the sphere that has the given hole, not the volume of the material that was removed.

http://www.stumblingrobot.com/2015/08/11/volume-of-cylindrical-hole-removed-from-a-sphere/

This website has the same answer as the back of the book. The math is there for your convenience. Thank you!

The answer given at the back of the book is [itex] (\frac {32}{3} - 4\sqrt{3}){\pi}r^3[/itex]

To drill a hole completely through the sphere, the hole would have to have a length of 4r.

To get the answer in the back of the book, it requires setting the length of the hole to [itex] r = \sqrt{4r^2-x^2} [/itex] and solving for r. So the hole has a radius of r, and a length of [itex]2\sqrt3r[/itex]

But this doesn't account for the material of the sphere that is greater than [itex]x=\sqrt3r[/itex] and less than [itex]x=-\sqrt3r[/itex]

Not only that, but the answer seems to be the volume of the sphere - the volume of a hole that doesn't completely go through the sphere. The question says "what volume of material is removed," The subtraction mentioned previously give you the volume of the sphere that has the given hole, not the volume of the material that was removed.

http://www.stumblingrobot.com/2015/08/11/volume-of-cylindrical-hole-removed-from-a-sphere/

This website has the same answer as the back of the book. The math is there for your convenience. Thank you!

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