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.