precalc prob?

Question

a cylinder is inscribed inside a sphere with radius R. let x=height of cylinder. write a formila for the volume of the cylinder as a function of x and that includes R in some way. What is the domain of V(x) and graph V(x) when R=3

Answer 1

The formula for the cylinder is pi*r^2*h. We know "h" or "x", but we have to find r.

If we draw a sectional view of the cylinder inside the sphere on a great circle of the sphere, we will see the cylinder as a rectangle, which we can call (from the lower left, clockwise) ABCD. We have the center of the sphere as point O. We draw a horizontal line OE to the left, where E is on the sphere, and on OE is a point F on the cylinder. Then OF is the radius of the cylinder. Also, R= OB and x/2= FB. .....
Consider angle FOB, which has the value of
sin-1 (x/2R). Since angle FBO is complimentary to FOB, it has the value 90-[sin-1(x/2R)]. By the law of sines, we can find OF:
........sin angle FOB/(x/2R) = sin angle FBO/OF.
Now you have r in terms of R.

As for part 2, x can range between 2R and 0. However, at both extremes, the volume of the cylinder is zero (at x -> 2R, r->0)