Chapter Three: Section Seven Optimization Problems

Chapter Three: Section Seven After spending a great deal of energy and time on sketching curves, we will turn our attention and energy to applying the ideas of finding Maximum and minimum values of a function to application problems. For our first example, we will return to that Precalculus favorite, the ‘Box Problem’.

Chapter Three: Section Seven Surely you remember this famous problem where you have a rectangle and you want to form it into a box by cutting out squares from the corners of the sheet to allow you to fold up the sheet to form an open-top box. You do remember these problems, right?

Chapter Three: Section Seven With the skills that we had in Precalculus, all we could do was to write the volume equation of the box that was created and look at graphs of the functions to discuss Maximum or minimum values for the function. Of course, we can now do this analytically rather than rely on our graphing calculators.

Chapter Three: Section Seven I want you to try one here;  A sheet of metal that is 24 feet by 20 feet is to be used to form an open-top box. We will do this by cutting out a square from each corner and folding up the sides to be welded to form our box. What should the dimension of this square be to Maximize the volume of the open-top box?  As usual, I want you to come to class prepared to discuss your efforts.

Chapter Three: Section Seven Now, let’s try a much more interesting problem using the same ideas of writing a function based on a verbal definition and then using the first derivative test to find local extrema.  A Norman window is one formed by a semi-circle atop a rectangle. Find the dimensions of such a window that has a Maximum area with a fixed perimeter of 16 feet.

Chapter Three: Section Seven We will be looking at quite a few more examples in class.