Applications of maximisation and minimisation

A step-by-step method for optimisation problems: choosing a variable, writing the quantity as a function of one variable, finding and testing stationary points, checking endpoints, and answering in context.

The method

Example 1: fencing beside a river●●

A farmer has $200$ m of fencing to make a rectangular paddock against a straight river, which needs no fence. Find the largest possible area.

Let each side at right angles to the river be $x$ m. The side parallel to the river is $200 - 2x$ m, with $0 < x < 100$.

$A = x(200 - 2x) = 200x - 2x^2$, so $A' = 200 - 4x$, which is zero at $x = 50$.

$A'' = -4 < 0$, so this is a maximum. $A = 50 \times 100 = 5000$.

The largest area is $5000$ m², with sides $50$ m and $100$ m.

Area as a function of the width

Drag X to try different widths. The area is largest at the vertex.

Try this

Why does the area become $0$ at both ends of the domain?

Example 2: the cheapest can●●●

A closed cylindrical can must hold $54\pi$ cm³. Find the radius that uses the least metal.

Volume: $\pi r^2 h = 54\pi$, so $h = \dfrac{54}{r^2}$.

Surface area: $S = 2\pi r^2 + 2\pi r h = 2\pi r^2 + \dfrac{108\pi}{r}$, for $r > 0$.

$S' = 4\pi r - \dfrac{108\pi}{r^2} = 0$ gives $r^3 = 27$, so $r = 3$.

$S'' = 4\pi + \dfrac{216\pi}{r^3} > 0$, so this is a minimum. Then $h = 6$: the best can is exactly as tall as it is wide.

Flashcards

First step in optimisation

Write the quantity as a function of one variable.

Justify a maximum

Second derivative negative, or a table of slopes $+, 0, -$.

Cheapest closed can

Height equals diameter.

Practice

With $200$ m of fencing on three sides of a rectangle beside a river, find the largest area in m².
In that problem, how long are the sides at right angles to the river, in m?
A closed can holds $54\pi$ cm³. Find the radius that minimises the surface area.
For that can, the height is:
Find the minimum value of $x + \dfrac{9}{x}$ for $x > 0$.
Two positive numbers add to $20$. Find their largest possible product.
Squares of side $x$ cm are cut from the corners of a $30$ cm square sheet to make an open box. Find the largest volume in cm³.
Why should you check the endpoints of the domain?
To show $x = 3$ gives a minimum of $S$, it is enough that: