More Optimization Problems

Main idea

It's the same method as before, just with trickier setups like cans, distances and travel times. Writing down the right function is the real problem; once you have it, the calculus is routine.

Example: a can of fixed volume

V = \pi r^2 h = 1000 \;\Longrightarrow\; A(r) = {\htmlData{sym=0}{2\pi r^2}} + 2\pi r h = {\htmlData{sym=0}{2\pi r^2}} + {\htmlData{sym=1}{\frac{2000}{r}}}

What the symbols mean

$2\pi r^2$
Top and bottom
Two circles of radius r.
$\frac{2000}{r}$
Side
The side area 2πrh after using the volume to replace h.

Explanation

These are more optimization problems: finding the best design or route, just with setups that take a bit more thought. All we're really doing is the same thing as last lesson. Take the can formula above. It looks busy, but it's only the can's surface split into pieces: $2\pi r^2$ (the top and bottom, two circles) and $2\pi rh$ (the side, which unrolls into a rectangle). The volume constraint $\pi r^2 h = 1000$ lets you swap $h$ for $\frac{1000}{\pi r^2}$ , and the side becomes $\frac{2000}{r}$ . Now it's a one-variable function again, and you know what to do with those.

The three setups below come up again and again. Each has a small trick that makes the algebra easier.

Cylinders. Volume $\pi r^2 h$ , surface area $2\pi r^2 + 2\pi rh$ . One of them is fixed, so use it to eliminate $h$ .

Distances. Minimize the squared distance instead of the distance. Think about why that's allowed: distance is never negative, and squaring keeps the order of nonnegative numbers, so whatever makes $D^2$ smallest also makes $D$ smallest. You get the same answer with no square roots to differentiate.

Travel time. Time is distance over speed for each leg; add them up. These problems explain why light bends when it enters water: it takes the fastest path (Snell's law). There's an animation further down that shows this.

As always, check that the critical point really is the optimum, by comparing with the endpoints or using the second derivative.

See it

Light travels from A in the air to B under the water, where it moves 25% slower. Slide the crossing point and notice the travel time: it's smallest, about 7.94, at x ≈ 3.59, past the halfway mark, because the light saves time by spending less of its path in the slow water. Setting T′(x) = 0 at that point gives Snell's law of refraction.

Worked examples

Example 1

A can must hold 1000 cm³. What radius uses the least material?

1
Using the volume to replace $h$ , the material is $A(r) = 2\pi r^2 + \frac{2000}{r}$ . Set $A'(r) = 4\pi r - \frac{2000}{r^2} = 0$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

For the 1000 cm³ can, find the optimal radius (in cm, to 3 decimals).

Problem 2+5 XP

For that can, what is the ratio $h/r$ ?

Problem 3+5 XP

A rectangle has its base on the x-axis and top corners on $y = 12 - x^2$ . What is its largest area?

Problem 4+5 XP

What is the shortest distance from the origin to the line $y = 2x + 1$ ?

Problem 5+5 XP

You're in a boat 3 km offshore, heading to a point 6 km down the shore. You row at 2 km/h and walk at 4 km/h. How far down the shore should you land?

Problem 6+5 XP

An open-top box with a square base must hold $32{,}000$ cm³. What's the least material (total area of base and sides) it can be made from?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A cargo ship's fuel costs $0.004v^3$ dollars per hour at $v$ km/h, and its crew and other fixed costs are $\$ 1000 $per hour. What speed minimizes the total cost of a$ 600$ km trip?

Where it's used

Light refracting at the surface of water, a lifeguard's best route to a swimmer, and the dimensions of a soda can (whose real proportions are close to this optimum, adjusted for the thicker lids) are all "more optimization" problems.

Light Takes the Fastest PathCalc I · Physics Getting the Most Power from a Solar PanelCalc I · Electrical Engineering

Optimization L'Hôpital's Rule and Indeterminate Forms