Question

A hemisphere of radius 3 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.

Answer 1

if heard langrange multipliers are good for this

Answer 2

or it has something to do with this kind of picture

Answer 3

v = pi r^2 h

r = 3cos(t) ; h = 3sin(t)

v = pi (3cos(t))^2 3sin(t) might be useful

Answer 4

v' = 27pi(-2cos(t)sin^2(t) + cos^3(t))

if i see it right

Answer 5

cos(t)(-2sin^2(t)+cos^2(t)) = 0 when t = pi/2, -pi/t which are impractical

-2sin^2(t)+cos^2(t) = 0 will be the only things that might make sense

Answer 6

-2s^2+c^2 = 0

c^2 = 2s^2

1/2 = tan^2(t)

1/sqrt(2) = tan(t); t = arctan(1/sqrt(2))