The Mathematics Education Department
EMT 725, Lisa F. Garey
Click here for the Problem statement
This problem of maximizing the volume of a cone can be investigated several ways.
A GSP sketch shows how the volume of the cone varies as ø varies.
The GSP sketch gives a maximum volume of 0.405 cubic inches when ø = 1.165 rad and R=1 in.
A spreadsheet yields similar values: a maximum volume of 0.4031 cubic inches when ø=1.15 rad with R=1.
The following solution utilizes the A.M.-G.M. inequality
C is the portion of the circle that forms the base of the cone with
radius r. Hence, the circumference of the base of the cone is 2*pi*r = C.
Setting up equations results in an expression for r in terms of the given
R and ø.
Volume of cone = V=
where h = height of cone
By Pythagorean Theorem, 
Substituting for r and h, we get 
By the A.M.-G.M. inequality, V = 
[Note: This use of the A.M.-G.M. inequality does not give us a function on the Right Hand Side of the inequality that is a constant. Therefore, even though we can find a value of x for which the two functions are tangent at that point, we can not argue that the function on the RHS is larger for all values of x. Note that the solution obtained below is not consistent with the ones reported earlier using GSP and Excel.]
with equality iff 
Substituting x into r, 
. This gives ø = 1.84 radians with
the maximum volume being 0.37 when R=1 in.
Return to Lisa's EMT 725 Problems