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You want to construct a closed box with a volume of 12 $\text{m}^{3}$ . The material for its top costs $\$4/\text{m}^{2}$ and the material for the bottom and sides costs $\$2/\text{m}^{2}$ . What are the dimensions that minimize the total cost of the materials?

Answer: base 2 m by 2 m, height 3 m

What is the maximum possible volume of a rectangular box inscribed in a hemisphere of radius R? Assume that one face of the box lies in the base of the hemisphere.

Answer: $V=(4/9)R^{3}\sqrt{3}$

A window is to have the shape of a rectangle surmounted by an isosceles triangle with horizontal base. The perimeter of the window is to be 8 m. What are the dimensions of the window such that it will have the greatest area?

Answer: 4.29 $\text{m}^{2}$

You must divide a lump of putty of fixed volume V into three or fewer pieces and form the pieces into cubes. How should you do this to maximize the total surface area?

Answer: three equal cubes

You must divide a lump of putty of fixed volume V into three or fewer pieces and form the pieces into cubes. How should you do this to minimize the total surface area?

Answer: one cube

What is the maximum possible volume of a rectangular box whose longest diagonal has fixed length L?

Answer: $3L^{3}\sqrt{3}$

A house in the form of a box is to have a volume of 10,000 $\text{ft}^{3}$ . The walls admit heat at the rate of 5 $\text{units}/\text{min}/\text{ft}^{3}$ , the roof at 3 $\text{units}/\text{min}/\text{ft}^{3}$ , and the floor at 1 $\text{unit}/\text{min}/\text{ft}^{3}$ . What should be the shape of the house to minimize the rate at which heat enters?

Answer: square base of side 10(5^2/3), height 4(5^2/3).

Find the ellipsoid with equation

$\begin{displaymath} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1\end{displaymath}$

that passes through the point (2,1,3) and has minimal volume, noting that $V=(4/3)\pi abc$ .

Answer: $\frac{x^{2}}{12}+\frac{y^{2}}{3} + \frac{z^{2}}{27} = 1$

About this document ...

This document was generated using the LaTeX2HTML translator Version 97.1 (release) (July 13th, 1997)

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latex2html -link 0 -split 0 -t Maximum and Minimum Word Problems maxmin.tex.

The translation was initiated by Joshua Holden on 11/10/1999

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Joshua Holden
11/10/1999