This is actually a really hard problem – it’s a really neat problem for students of calculus, but it’s hard, like 4 or 5 stars hard. This puzzle is also a real problem, meaning it actually saves someone some hassle or energy or time or money. As with many word problems, if you imagine yourself in the situation, or on the hook for the outcome, you might find it easier.
It was first encountered by Welches, the grape jelly manufacturer, while making their primary product, grape jelly.
Standard jelly- or jam-making means cooking fruit juice to drive off the water and make it more concentrated. But heating is expensive, so naturally people sought low-temperature ways, and the easy way is evaporation by moving more air, and exposing a larger surface to the air.
One easy way to get more jelly to dry out is to spin a wheel in it, so that the jelly is lifted out of the pan on the wheel, increasing the surface area exposed to the air.
If you only barely dip the edge in, only the circumference of the wheel gets wet, and only the edge really works to increase the surface area of the jelly exposed to air, although it is almost the entire edge – all but the tiny bit dipped, which is protected by the liquid.
If you dip the wheel half way, so that the axle itself is in the liquid, too, then half the wheel is under the surface of the liquid, which doesn’t expose anything to air, although then you do get the entire top half of the wheel wet, which means at least half is exposed to air.
Of course one obvious way to increase the wet exposed surface is to have lots of wheels, which of course they all do, now. But it’s still worthwhile to make each wheel maximally efficient.
If we set the wheel at exactly the right height, then we can maximize the amount of surface of the exposed, wet wheel.
So what is the proper distance of the axle above the surface of the liquid, to maximize the wet, exposed surface?
The standard way to answer is in terms of a fraction of a radius as the picture suggests.