Item 17 a sphere and a right cylinder have the same radius and volume. find the radius rr in terms of the height hh of the cylinder.

Sphere and right cyl. have the same radius and volume.  Thus, find the equations for the volumes of a sphere and a right cylinder and set them equal to one another:

Vol. of sphere       =       Vol. of right cyl.

  (4/3) pi r^3          =        pi r^2 * h

"pi" appears on both sides of this equation as "pi."  So we can cancel "pi" from both sides.  We then have:

(4/3) r^3  =  r^2 h

Let's now divide both sides of this new equation by r^2.  We then have:

(4/3) r = h

We must solve this for r in terms of h.  To do this, multiply both sides of this equation by (3/4).  This results in:

r = (3/4) h     This is the final  the radius, r, in terms of the height, h.