# Optimization problems

## Maximum Volume

With the intuition your students developed while experimenting with derivatives, they are prepared to tackle a simple optimization problem:

An ancient windmill is to be restored and made into a restaurant. The owner of the windmill needs to integrate a cylindric water tank for the restaurant's kitchen and restrooms in the cone shaped roof of the windmill. What size should the water tank be to hold a maximum amount of water?

In the dynamic construction below you can investigate the problem by dragging the red point with the mouse to change the radius `r` of the cylinder.

Editor's note, May 2014: For an HTML5 version of the above applet,
click here.

Additionally, this construction allows you to explore whether the volume of the inscribed cylinder *always* has a maximum, independent of the measures of the cone. For this purpose you should first right click (MacOS: apple click) on point `V` and deactivate its trace. Afterwards you can create the locus line of point `V` by typing the command `Locus[V, R]`

into the input field and hitting *Enter*. When you drag one of the brown points to vary the size of the cone, the locus line will adapt automatically. Therefore you are able to examine the effect on the maximum volume of the cylinder.

## Maximum Surface

To show that similar extremum problems don't always have a solution, we prepared another mathlet in which we replaced the parameter "volume of the cylinder" by "surface of the cylinder". In this example, the surface of the cylinder does not always have a maximum value. Change the size of the cone in the construction below to find out why.

Editor's note, May 2014: For an HTML5 version of the above applet,
click here.