what does the volume of a cylinder equal to?

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matherapist

Open Questions

what does the volume of a cylinder equal to?

A cylinder of base radius R and Height 2R has a volume that is equal to the sum of the volumes of two other dissimilar solids.
What are they and what are their dimensions?
Hint: this problem was solved by Archimedes many years ago.

Posted 157 day ago

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Answers (7)

math_fanatic, please note the following constructions in geometry are impossible.

1) dividing a random angle in three equal parts. This is known as TRISECTING AN ANGLE. Even today, many people claim that they can do it. But mathematically it has been proved that using unmarked straightedge and a compass, it cannot be done.

2) drawing a square that has the area of a given circle. This is known as squaring a circle which is impossible..

3) Doubling the volume of a given cube.

In your posted solution, you tried to square a circle that is why you tried to do something that mathematically has been proven to be impossible.

Just remember these impossible construction for your future solutions.
You may ask, if we need to trisect an angle, what should we do? Nothing! Just do it, but your divided angles are not exactly equal, they may differ by a tiny little bit of a degree, practically unimportant at all.
In practice, we do a lot of things that strict mathematically are not possible! such as doubling the volume of a given cube.
Thank you for you time reading my comments. rajbaba78.

Posted 156 day ago

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matherapist

Math-fanatic- Your solution is interesting, however since PI is irrational, having a side of a cube equal to a multiple of it or a root does not seem correct. Yes, the circumference of a circle is a multiple of PI, however circles and spheres are the only figures where PI is used as a descriptor for volume or area or length.

You may want to try finding how many spheres of radius R/2 will equal the volume of the given cylinder above.

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math_fanatic

Now I've looked at Raj's answer.
It's obviously prettier than mine, but mine still works.

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math_fanatic

I know other people have answered this question but I was careful not to look at anyone's answers to see if I could come up with it on my own. I don't know if I'm doing something wrong, but this problem seems particularly easy. Perhaps, Archimedes did not have the benefit of knowing all the formulas for the volumes of solids like we do.

Note: I don't know how to make the symbol for pi with my keyboard so I'm just spelling it out.

The two solids are:
1) a sphere with a radius of r
2) a cube with each side equal to [(2/3pi)^(1/3)]r

the volume of a sphere with radius r has a volume of 4/3pi r^3
the volume of a cube with side [(2/3pi)^(1/3)]r has a volume of 2/3pi r^3

You add them together to get 2pi r^3 which is the volume of a cylinder with radius r and height 2r

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matherapist

Professor rajbaba78: We (tutors) all know you can answer anything we ask. These problems are for students /tutors who have the desire to try to solve them. In the future, please give them helpful hints to solve the problem, and I will do the same for your problems. Knowing the answer and writing it down does not really prove anything, especially if you have seen it . If you looked it up and understood it, that's OK but admit what you did.

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samzappala

r^2*3.14= base
2r= height
r^2*3.14*2r=vol=r^2*3.14*1.5r+r^2*3.14*.5r
After simplifying, r (original)=dissimilar r1+dissimilar r2. In this case, 2r=1.5r+.5r. Since there is not a value, all I can give you is the formula (above). Any figure can be used as long as r=r1+r2.

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The volume of the cylinder is base area times it's height, so it is:

v = (PI)R^2.(2R) = 2(PI)R^3. This is equal to a sphere + a cone with the following dimentions:

1) Sphere with radius R, V1 = (4/3)(PI)R^3

2) Cone with circular base and radius R and height 2R:

Volume of a cone = base area times (h/3)

V2 = (PI)R^2.(2R/3) = (2/3)(PI)R^3

Total of (V1 + v2) = [(4/3) + (2/3)] (PI) R^3 = 2(PI)R^3

i.e. v = v1 + v2 <== answer

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