A hemisphere of radius 1 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of?

Question

(rest of question) 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.

(rest of question) 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.

Now, I know it's hard to visualize, but that is all that is given.

Answer 1

Inscribe a cylinder in a hemisphere. Find the dimensions of the the cylinder with maximum volume in terms of the radius of the sphere.

r = radius sphere
x = radius cylinder
y = height cylinder
V = volume cylinder

V = πx²y
x² = r² - y²
x =√(r² - y²)

Substitute the value of x² into the formula for V.

V = π(r² - y²)y = π(r²y - y³)

Take the derivative and set equal to zero to determine critical values.

dV/dy = π(r² - 3y²) = 0
r² - 3y² = 0
3y² = r²
y² = r²/3
y = r/√3
Distance can't be negative.

Take the second derivative to determine the nature of the critical point.

d²V/dy² = -6πy < 0
This implies a relative maximum which is what we want.

y = r/√3
x² = r² - y² = r² - (r/√3)² = r² - r²/3 = 2r²/3
x = r√(2/3)

radius cylinder x = r√(2/3)
height cylinder y = r/√3

For a hemisphere of radius 1 this reduces to:

radius cylinder x = √(2/3)
height cylinder y = 1/√3

Answer 2

I assume the hemisphere has its flat side on the plane, and that the cylinder is inside the hemisphere.

The best way of tackling this is to draw it from the side, as a half circle sitting on a line, and a rectangle representing the cylinder. The point where the cylinder touches the circle represents the circular rim of the cylinder.

You know that the height of the cylinder is related to the radius of the cylinder, because they both lie on the circle, so r² + h² = 1.
So r² = 1 - h²

Calculate the volume of the cylinder:

V = Pi.r².h
= Pi.(1-h²)h

Now using calculus, maximise this by appropiate choice of h.

Answer 3

I can't visualise this. Is the hemisphere lying with its flat side on the plane or balancing on a point? How can the top of the cylinder touch the hemisphere? Is it above it or below it or what?

Edit. I naturally thought that the hemisphere was solid. The question could have been worded better. I won't continue with the how to calculate the answer as others have done so.