When last we left our intrepid father, it was a time of night he hadn’t seen since college and he was standing outside in boxer shorts and gooseflesh waiting for the family dog to finish up so he could crawl back to bed. But what’s the chance said family dog would fertilize the neighbor’s prize orchid without flattening it?
Here’s the cliff notes version: your hand is at the edge of a sidewalk, and so your dog’s 10-foot leash is (almost) the radius of a perfect semicircle from that point — only, your hand is held 3-feet high and the dog’s collar is at 1 foot high. Somewhere in this semicircle is an orchid whose roots extend 16 inches in all directions and a 4-inch square kill zone near its stalk. What are the chances your dog will fertilize the neighbor’s orchid without killing it?
• First figure out the area of your dog’s semicircular reach. Sure, the leash is 10-feet long, but because the dog is lower than your hand, 10 feet is not the radius of this semicircle. Instead, 10 feet is the hypotenuse of a right triangle with a short side 2 feet. So: 2^2+n^2=100. And the radius of the semicircle is 9.8 feet plus the 30 inches your dog can “hang it” for a total of 147.6 inches and an area of .5*πr^2 or 34,221 square inches.
• Now calculate the “sweet spot” — the total area of the space in which your dog hits the roots without hitting the stalk. This is fairly easy: it’s a circle with 16″ radius minus a square with 4″ sides. So: π16^2-4^2 = 788 square inches.
• Now to probability! There’s a “788 in 34,221” chance or a 0.023 chance (or a 2.3% chance) that your dog will fertilize the orchid without killing it.
There were quite a few correct solutions (and also a handful of incorrect solutions…you know who you are…). First, thanks to Brooks, Rich, and Brad for their spectacularly illustrated explanations. I headed the post with Brooks’ truly awesome work (please let me know if you’d prefer I take them down — I’m happy to pull ’em if you like). Though I’m sorry to say none of these radical overachievers was randomly selected as this week’s winner. The winner of the $50.00 ThinkGeek Gift Certificate is… drum roll please… Bryce! For the rest of you, this month’s ThinkGeek discount code is GEEKDAD59MJ.
Congrats to all who entered! Thanks for reading GeekDad and tune in Monday when the estimable Dave drops mad rhymes from the land of puzzling geekery, presumably without reference to feces.