Determine the radius and height that maximizes the volume of a cylinder which fits within a cone with a radius of 5 cm and a height of 12 cm.

Question

What is the maximum volume of the cylinder? Can you generalize a formula for a cone of radius r and height h?

Answer 1

Start by drawing a picture---you have a cylinder inside of a cone. At one end of the cylinder, the diameter is the same as the cone.

With the wide end of the cone down, we have the following relationships:

Volume of the cylinder = h * pi * r²

Radius of the cylinder at the height where it intersects the cone: r = (12-h)* 5/12

Substitute this equation for r into the equation for the volume of the cylinder:

V = h * pi * r²

V = h * pi * (12-h)* 5/12)²

Expand and Simplify:

V = h * pi * (5 - 5h/12)²

V = h * pi * (25 - 25h/6 + 25h²/144)

V = pi * 25 (h - h²/6 + h³/144)

V = pi *25*144 (h³ - 24h² + 144h)

To find the maximum, take the 1st derivative and set = 0

V' = pi *25*144 (3h² - 48h + 144) = 0

h² - 16h + 48 = 0

h = 4, 12

Recognize that h = 12 is the upper limit, where the volume of the cylinder will be 0. The maximum occurs at h = 4.

To visualize all this---and to check it, use a graphing calculator or online tool to plot the equation.