Chapter 7 Differential Calculus

Section 7.5 Applications

7.5.4 Optimisation Problems

In many applications in engineering and business, solutions to problems can be found which are not unique. They often depend on variable conditions. To find an ideal solution, additional properties (constraints) are defined that are to be satisfied by the solution. This very often results in so-called optimisation problems, in which one solution has to be selected from a family of solutions such that it best satisfies a previously specified property.
As an example, we consider the problem of constructing a cylindrical can. This can must satisfy the additional condition of having a capacity (volume) $V$ of one litre (a.k.a. one cubic decimetre, $1\hspace{0.17em}\mathrm{dm}{}^3$). Thus, if $V$ is specified in $\mathrm{dm}{}^3$ and $r$ is the radius and $h$ the height of the can, each measured in decimetre ($\mathrm{dm}$), then the volume is $V=\pi r^2·h=1$. The can with the least surface area $O=2·\pi r^2+2\pi rh$ is required in order to save material. Here, the surface area $O$, measured in square decimetres ($\mathrm{dm}{}^2$), is a function of the radius $r$ and the height $h$ of the can.
In mathematical terms, our question results in the problem of finding a minimum of the surface function $O$, where the minimum has to be found for values of $r$ and $h$ that also satisfy the additional condition for the volume: $V=\pi r^2·h=1$. In the context of finding extrema, such an additional condition is also called a constraint.