Learning to solve for volumes? help appreciated.

[FrostCS]

Hello Everyone,
I am a bit new here, and I hope this will be the correct area to ask for help on this. (I've been away from math for a good number of years).

I am trying to figure out how to find the volume of an area removed from a cylinder.

Say I have a cylinder with a radius of 6 units, and a height of 5 units standing on a table. I have a volume of 565.7142857142852 cubic units or so..

But, I'd like to drill a hole into the side of this cylinder using a drill bit with a radius of 1.5 units (3 unit diameter). This would create a cylinder of empty space inside it, except the ends of the empty-cylinder would be rounded to the shape of the orginal cylinder, right?

How would I go about finding the volume removed?


I've been searching for this for a few days now.. I can find arc segment length, segment width.. obviously the radius is 6 units, and also end up with an arc height.. but there has to be some way to figure out the volume of the rounded ends to the cylinder, right?

Any help is much appreciated, I am in no rush to find an answer, but it's been stuck in my mind for a while now :-).

Regards,
C. Frost

 

[cristo]

Ok, this isn't a mathematical answer, but do you know the density of the substance the cylinder is made of? If so, you can measure the mass after drilling a hole and calculate its volume.

 

[FrostCS]

Ok, this isn't a mathematical answer, but do you know the density of the substance the cylinder is made of? If so, you can measure the mass after drilling a hole and calculate its volume.

This would be an option, except I was hoping to know the volume before drilling, which would be easiest if I was able to calculate it without drilling :-) The alternative would be to drill and dunk it in a container of water, and measure the difference before and after drilling :-) There just has to be a mathematical way to figure out the volume, without drilling, right?

I can get the volume of the cylinder to be drilled using the apothem, and the radius, just not those rounded ends to the cylinder...

 

[mathwonk]

slice the cylinder by a vertical plane. notice it makes a rectangular slice on the cylinder. can you see what shape the drill cuts from this rectangle?

if you can compute the area of this shape, then you can also compute the volume of the drill hole.

just a suggestion.

or maybe slice the cylinder by a horizontal plane, notice it slices the cylinder in a circle, and ask what shape the drill hole cuts from this circle.

gosh that looks easier,a s it seems to be a rectangle.

anyway, just try to reduce the problem of finding the volume of the hole, to finding the area of a section of the hole. then move the section and integrate.

 

[FrostCS]

slice the cylinder by a vertical plane. notice it makes a rectangular slice on the cylinder. can you see what shape the drill cuts from this rectangle?

if you can compute the area of this shape, then you can also compute the volume of the drill hole.

just a suggestion.

or maybe slice the cylinder by a horizontal plane, notice it slices the cylinder in a circle, and ask what shape the drill hole cuts from this circle.

gosh that looks easier,a s it seems to be a rectangle.

anyway, just try to reduce the problem of finding the volume of the hole, to finding the area of a section of the hole. then move the section and integrate.

Maybe my question isn't worded so well, What I am doing is taking Cylinder A, and intersecting it at 90 degrees by drilling Cylinder B into it. The hole isn't being drilled into the cylinder like on a flat surface, but drilling into the rounded surface on the side of one cylinder. Picture one cylinder laying on it's side, and the other one standing upright.

This way, if I slice it horizontally, I don't get a rectangle... I do get a circle if I slice it vertically, but then I still end up with the shape at the end that I can't solve for.

I believe this shape is called a "plano-convex" lens, yet I can't find how to solve for the volume of this object. If I had the volume of this object, then I could find the end area.. but at this point I don't know how to solve for the end area either.

 

[hp48gx]

Is this a physics forum or what? ;) This is a tough problem and I have the answer for you. It requires a triple integral if you want "exact" but if you can suffer being 1 or possibly 2 cubic inches off (depending on what you are cutting), you can get really close (1 or 2 cubic inches above).

I may just work through both of them because I have been working on this for months and finally gave in and emailed my calculus teacher from college.

My cylinder is 48" I.D. and 58" O.D. making for a 5" thick wall. I am cutting a 24" diameter hole through one side of the cylinder (as you described).

So, do I need to go on?