Proving the equation for the height of a cylinder

[Mohamed Abdul]

Homework Statement

Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed.
Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the band.

Homework Equations

V=pi*r^2*h

The Attempt at a Solution

I've finished all the necessary integrals and arrived at 4/3*pi*(a^2-b^2)^3/2. The answer in the textbook states that (a^2-b^2)^3/2=h^3. so that'd mean that sqrt(a^2-b^2) = h. I don't know how to prove that relationship, however, so that is my biggest problem with this question.

 

[LCKurtz]

What is $b$?

 

[Mohamed Abdul]

What is $b$?

b is the radius of the cylinder which I am cutting out of the sphere, sorry that I didn't explain that earlier.

 

[LCKurtz]

Your original statement of the problem says the radius of the cylinder is $a$.

 

[Mohamed Abdul]

Your original statement of the problem says the radius of the cylinder is $a$.

I set b to be the lower boundary of r. So the final value should have really said: 4/3*pi*(A^2-a^2)^3/2, where A is the radius of the sphere and a is the cylinder radius. I just need to know how (A^2-a^2)^3/2 = h^3

 

[LCKurtz]

It helps to be clear what the variables are. In your OP you stated the volume of the cylinder is $\pi r^2 h$ which implies you are using $h$ as the height of the cylinder. If that is the case, then $\frac h 2 =\sqrt{A^2-a^2}$. But if you are using $h$ for half the height of the cylinder, then $h =\sqrt{A^2-a^2}$ which would give you your formula.

 

[Mohamed Abdul]

It helps to be clear what the variables are. In your OP you stated the volume of the cylinder is $\pi r^2 h$ which implies you are using $h$ as the height of the cylinder. If that is the case, then $\frac h 2 =\sqrt{A^2-a^2}$. But if you are using $h$ for half the height of the cylinder, then $h =\sqrt{A^2-a^2}$ which would give you your formula.

Wait what's the equation where you got that h equals that?Do you set the volume of the cylinder equal to something?

 

[LCKurtz]

Wait what's the equation where you got that h equals that? Do you set the volume of the cylinder equal to something?

Draw a picture of the cross section of your figure and label radii and height in the figure.There's an obvious right triangle in there.

 

[Mohamed Abdul]

Draw a picture of the cross section of your figure and label radii and height in the figure.There's an obvious right triangle in there.

I'm drawing the picture but I'm not seeing anything. I set the triangle up along the edge of the sphere and got a base of A-a, but I'm not sure of the angles to find the other side lengths.

 

[LCKurtz]

Here's a picture where $h$ is half the height of the cylinder:

 

[Mohamed Abdul]

Here's a picture where $h$ is half the height of the cylinder:
View attachment 213500

Thank you, I understand that half the height would equal the square root of the square of the distances between A and a