Question

Help with minimum surface area of cone given volume

Answer 1

A cone shaped cup is to hold 27cm^3 water. What is the height and radius to use the least possible materials. This is what I've done so far, but something's wrong!

Answer 2

Surface Area (given open top):\[SA = \pi r h\]
Volume:
\[27 = \frac{ 1 }{ 3 }\ base area (r^2)\]
Base Area:
\[A = \pi r^2\]
Base area into Volume:
\[27 = \frac{ 1 }{ 3 }\pi r^2 h\]
So:\[h = \frac{ 81 }{ \pi r^2 }\]
Putting that into Surface Area:
\[SA = \pi r \frac{ 81 }{ \pi r^2 }\]
\[ = \frac{ 81 }{ r }\]

Answer 3

But taking the derivative of SA:
\[SA' = \frac{ -81 }{ r^2 }\]
And since SA is at a min/max when SA' = 0:
\[r^2 = -81\]
And then:
\[r = 9i\]
Which doesn't work in the question. HELP! Where did I go wrong?