A cylinder is inscribed in a right circular cone of height 28 and radius (at the base) equal to 8. What are the
dimensions of such a cylinder which has maximum volume? The volume of a cylinder is $V = \pi r^2h$
[A] Find a function for the volume in terms of the radius of the cylinder r. Use r as the variable.
V(r) =
[B] Find the appropriate domain of V in the context of the problem. Use interval notation.
Domain:
[C] Find V'(r):
V'(r) =
[D] Find the critical value(s) within the appropriate domain of V. No decimal entries.
Critical value(s):
[E] Determine where V is increasing and where V is decreasing on its appropriate domain.
Increasing:
Decreasing:
[F] What is the best conclusion with regards to an absolute extrema at the critical value.
$\circ$ Since the V goes from increasing to decreasing, by the first derivative test we have an absolute
maximum at the critical value
$\circ$ Since the V goes from increasing to decreasing, by the first derivative test we have an absolute
minimum at the critical value
$\circ$ Since the V goes from decreasing to increasing, by the first derivative test we have an absolute
minimum at the critical value
$\circ$ The results are inclusive, another test besides the first derivative needs to be performed in order to
determine if there is an absolute maximum or minimum at the critical value
$\circ$ Since the V goes from decreasing to decreasing, by the first derivative test we have an absolute
maximum at the critical value
[G] What are the dimensions of the cylinder that maximizes its volume
Radius =
Height =
[H] Find the maximum volume of the cylinder that can be inscribed in the cone.
Maximum Volume =
