Chapter 5 section 4

Excel Companion Chapter 5 section 4

ClsscMin.xls

One of the best known classical optimization problems involves finding the dimensions ( radius and height) of a cylinder with a fixed volume which has the minimum surface area.

Clsscmin.xls - min area (cylinder)

Questions

9.	Let the independent variable r represent the radius and let V be the constant volume. Explain how to obtain the height of the cylinder in terms of the volume and radius and also how to obtain a function A(r) for the surface area of the cylinder in terms of the parameter V and the variable r.
a)	Determine the first derivative of A(r)
b)	Determine the second derivative of A(r)
c)	Find the value of r that minimizes A(r).
d)	What is the ratio of the height to the radius for the optimal solution?. How does this compare with the soup cans in the supermarket?
e)	Apply Solver to min area (cylinder) in ClsscMin.xls using different volumes and compare these results to the optimal solutions predicted by calculus

Situation: A postal service imposes a restriction on packages that they will handle for a standard rate. The length plus the perimeter of a cross section (girth) can be no more than 90 inches (there is a high surcharge for packages that do not meet this restriction). A box manufacturer gets an order to construct shipping cartons that will have the maximum volume, meet the postal restriction, and also satisfy the condition that one dimension of the rectangular cross section be twice the other dimension.

10.a)	Construct a function for the volume of the box
b)	Write the formula for the volume function in terms of a single variable.
	Develop your own spreadsheet showing a table and grap-h of this function. Save it as Post**.xls where each is one of your initials.
c)	Find the optimal solution using Solver or calculus.

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