Volume of a cylinder is thrice the volume of a cube?

Question

you have to prove that the Volume of a cylinder is thrice the volume of a cube

Answer 1

That's not something that can be proven in general, because it's not true in general.

If the question were, for instance, "What would be the length of the side of a cube that is 1/3 the volume of a given cylinder with height h and radius r?", that would be answerable. (Three times pi times r squared times h, the whole thing to the 1/3 power.)

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Ya know, after thinking about it, I am betting that your question should read:

"Prove that the volume of a cylinder is thrice the volume of a CONE with the same radius and height."

THAT is a true statement, and there are proofs of it is here:

http://mathforum.org/library/drmath/view/53646.html

and here:

http://www.mph.net/coelsner/calcapps/cone_ex.htm

Answer 2

Assuming the pyramid must be a regular square pyramid, The largest square pyramid inside a cube with side length d, will have height f = d and base side e = d (such that it shares a cube's side as its base), and its apex at the center of the opposite cube side, such that its volume V[pyramid] = (1/3) e^2 f = d^3 / 3 = one-third the volume V[cube] of the cube. Thus, V[pyramid] is maximized as V[cube] is maximized. A cube inside a cylinder with radius b and height c, will have volume V[cube] such that: V[cube] = min(2b, c)^3 *** Eq. 1 Since c is inversely proportional Assuming both caps of the cylinder are such that their circular perimeters are parallel "small circles" of the sphere, (that is, for the cylinders maximum radius b[max] = 1, c = 0; and likewise, for the cylinder's maximum height c[max] = 2, b = 0). Let point o be the center of the sphere, and Let point p be on the perimeter of a circular cap. Let oq be a radius of the cylinder such that pq is perpendicular to the circular cap. Then, op = the spherical radius a = 1, pq = half the cylinder's height = c/2, and oq = the cylinder's radius b opq is a right triangle with hypotenuse op. By the Pythagorean Theorem, c^2 / 4 + b^2 = 1 => c = 2 sqrt(1 - b^2) *** Eq. 2 Since 0 < b < 1 and 0 < c < 2, we need only consider the positive branches of the sqrts. Now, b is inversely proportional to c, and both are monotonic, meaning that (by Eq. 1), V[cube] is maximized where b = c. So our Pythagorean equation can be simplified to: b^2 / 4 + b^2 = 1 => b = 2 sqrt(5) / 5 And, so V[pyramid, max] = V[cube, max] / 3 = b^3 / 3 = (2 sqrt(5) / 5)^3 / 3 = 8 sqrt(5) / 75 *** SOLUTION ≈ 5.7% of the volume of the sphere --- Since the inradius of a unit cube is 1/2 and a circumradius is sqrt(3)/2, I would imagine packing the sphere in the cube would maximize space utilization. The most awkward volume to fill seems to be the pyramid. So I would guess a maximal order would go like this (from innermost to outermost): pyramid, cylinder, cube, sphere I would guess that a minimal order would be cylinder, cube, sphere, pyramid

Answer 3

First of all, what u said is impossible. It's " Volume of cylinder=3*volume of cone". U can prove this by doing a simple experiment- Take a cylinder and cone of same radius and same height. Than fill the cylinder with water withe help of same cone( fill it full). U will observe that 3 times u fill the cone with water, then only cylinder gets full. Hence, my above expression is proved.

Answer 4

the volume of a cylinder is represented by the formula
p*r*r*h, where p is pi(22/7 or 3.14),r is the radius of the cylinder,h is the height of the cylinder.
the volume of cone is represented by the formula
1/3(p*r*r *h)where p is pi,r is the radius of the cone and h is the height of the cone.
(p*r*r*h)represents the volume of the cylinder.
therefore,volume of cone=1/3(volume of cylinder)
cross multiplying,we get
3*volume of cone=Volume of cylinder.
so,the volume of cylinder equals thrice the volume of cone

Answer 5

Let r,h be the radius and height of cylinder. Your question is a particular case.
Let edge of cube be CUBE ROOTof "PI *r^2*h"/3"=L
Now volume of cylinder =V=PI*r^2*h (1)
volume of cube =V1=L^3
=(CUBE ROOTof "PI *r^2*h"/3)^3
="PI *r^2*h"/3
=V/3 from (1)
implies 3V1=V or V=3V1

Answer 6

you mean for what dimensions?unit radius,unit height of the cylinde and unit edge of the cube?