optimization

[renegade05]

A right-circular cylinder is inscribed inside a cone of height 6 meters with base radius 4 meters. Find the largest possible volume of such a cylinder. Solve analytically using calculus. Include an appropriate test.

I came up with the equation \(\displaystyle 6\pi r^2 -3/2 \pi r^3 = volume\) r = radius of cylinder.

Is this equation correct?

All i have to find is the absolute max on interval (0, infinity) right?

and what would be an appropriate test?

 

[galactus]

You appear to have the correct volume. Good.

The variable r must satisfy \(\displaystyle 0\leq r\leq 4\)

Differentiate your volume formula.

Set to 0 and solve for r.

There will be two critical points. The max will occur at one of these points.

Sub these values into the volume formula you came up with to see which is the largest.

Actually, if you check, you will see that the volume of the cylinder is 4/9th that of the cone.

 

[renegade05]

The variable r must satisfy \(\displaystyle 0\leq r\leq 4\)

\(\displaystyle 0\leq r\leq 4\)
or
\(\displaystyle 0< r<4\)

??

 

[SigepBrandon]

galactus was correct.
I'm on the same section as you, and there was something in my notes about the interval having to be closed. It makes sense because you want to test the end points (for local max/min) and if the boundary is not actually included in the function then you can not test them. I'm sorry for the simple explanation, I'm sure someone else could explain it in more detail-
brandon