Find the Radius of a Cylinder Inside a Hemisphere with Multiple Integration

[nas]

if i have a hemisphere of radius 4, is it possible using multiple integration for me to find the radius of a cylinder that sits inside the hemisphere such that the vol inside the hemisphere and outside the cylinder is a 1/12 of the vol of the hemisphere
anyone that can help me on this-i respect you
my inital thoughts were do i do this via polar cordinates-but the hgt of the cylinder is causing me problems-or do i assume hgt is also the radius of the cylinder?

 

[matt grime]

I suppose so, but it seems very unneccesary.

let r be the radius of the cylinder - find it's height in terms of r (it touches the surface of the sphere presumably) You can now find the volume of the cylinder in terms of r. Find the r that satisfies the criterion you gave using simple algebra (you know the volume of the sphere too).

 

[M98Ranger]

Yes, it is possible to find the radius of a cylinder inside a hemisphere using multiple integration. This can be done by setting up a triple integral, where the innermost integral represents the radius of the cylinder, the middle integral represents the height of the cylinder, and the outermost integral represents the angle of rotation (using polar coordinates).

To solve for the radius of the cylinder, you would first need to find the volume of the hemisphere using a single integral. Then, you can set up the triple integral and use the given information of the volume of the hemisphere and the desired volume of the space between the cylinder and the hemisphere to solve for the radius of the cylinder.

It is important to note that the height of the cylinder may not necessarily be the same as the radius of the hemisphere. You may need to use a variable for the height and solve for it along with the radius of the cylinder.

I would also recommend consulting with a math tutor or professor for guidance on setting up the triple integral and solving for the radius of the cylinder. Best of luck!