This week the #puzzle is: Can You Irrigate the Garden? #probabilities #geometry #trigonometry #average #integration

You and your assistant are planning to irrigate a vast circular garden, which has a radius of 1 furlong. However, your assistant is somewhat lackadaisical when it comes to gardening. Their plan is to pick two random points on the circumference of the garden and run a hose straight between them.

You’re concerned that different parts of your garden—especially your prized peach tree at the very center—will be too far from the hose to be properly irrigated.

On average, how far can you expect the center of the garden to be from the nearest part of the hose?

And for extra credit:

As before, your assistant intends to pick two random points along the circumference of the garden and run a hose straight between them.

This time, you’ve decided to contribute to the madness yourself by picking a random point inside the garden to plant a second peach tree. On average, how far can you expect this point to be from the nearest part of the hose?

Can You Irrigate the Garden?

Highlight to reveal (possibly incorrect) solution:

Program Desmos

My first attempt was a monte carlo program.

My second was to use integration. I convert from “2 random points” to “1 random point and 1 fixed in a nice way” to “2 random points but with similar angles”. That way the integral is nicer.

Result: 2/π ≈ 0.636619772368.

And for extra credit:

Desmos

My first (and only) attempt was the program.

Note that the line going through the 2 hose points might be closer to the random point than the hose itself.

Result: 0.74178.