Volume With Disk Method

[jamesk]

I really need help understanding how to work these calculus problems. I know they involve using the disk method. I don't care as much about the answer as I do understanding how to get the answer. I really appreciate any help wha
tsoever.

1. A tank on a water towers is a shere of radius 50 feet. Determine the depths of the water wheb the tank is filled to 1/4 and 3/4 of its total capacity.

2. A manufacturer drills a hole through the center of a metal sphere of radius R. The hole has a radius r. Find the volume of the resulting ring. What value of r will produce a ring whose volume is exactly half the volume of the sphere?

Thanks again

 

[MarkFL]

1.) I think I would choose to derive a general formula for a spherical tank of radius \(\displaystyle R\), orient a vertical axis through the center of the tank with the origin at the center, compute the volume of an arbitrary slice, and integrate from \(\displaystyle -R\) to \(\displaystyle -R+h\) (where \(\displaystyle h\) is the depth of the water) and equate this to \(\displaystyle k\cdot\frac{4}{3}\pi R^2\) where \(\displaystyle 0\le k\le1\) is the portion of the talk occupied by water. Then solve for \(\displaystyle h\) in terms of \(\displaystyle k\) and \(\displaystyle R\), and plug in the given values of \(\displaystyle k\) and \(\displaystyle R\) at the end.

What is the volume of an arbitrary slice? This needs to be a function of your axis variable, which I would call \(\displaystyle x\).

 

[MarkFL]

After actually working the problem, I realize a numerical method is needed, unless one wishes to utilize the cubic formula, which I don't.