Right now I’m driving from Chicago to Salt Lake City to start a job at the University of Utah. My route has taken me past a lot of silos and water towers, which naturally make me think of related rates. (Sadly, none of them seemed to be leaking into polyhedral receptacles.)
A cylindrical silo in South Dakota. Image: flickr user Lars Plougmann.
For the most part, mathematical concepts tend to percolate into my brain slowly. But related rates came to me all at once. I remember finding them so impenetrable that I actually went to the math lab for help with my homework when I first encountered them. In the middle of one of the problems I was working on with the tutor, it just clicked. Suddenly I could do related rates, and I couldn’t figure out why I had found them challenging before. I keep hoping other difficult concepts will suddenly become clear to me in the same way, but alas, it hasn’t happened again.
A few weeks ago, I read a post on the blog Intersections–Poetry with Mathematics about related rates in fiction and poetry. JoAnne Growney, a poet and former math professor, shared two excerpts that play with the artificiality of word problems about related rates. The first is from Kyi May Kuang’s book Black Rice, and the second was from the poem “The Calculation” by David Wagoner. Growney observes that both authors seem to use the idea of a word problem about related rates to cultivate empathy with the audience. Wagoner’s poem begins like this:
“A man six feet tall stands on a curb, facing a light suspended fifteen feet above the middle of a street thirty feet wide. He begins to walk along the curb at five m.p.h. After he has been walking for ten seconds, at what rate is the length of his shadow increasing?
—a problem given by my calculus instructor, Penn State, 1946
Facing a streetlight under batty moths
And June bugs racheting like broken clock springs,
I stand, for the sake of a problem, on the curb—
Neither in grass nor gutter—while those wings
Switch down the light and patch my undershirt.
I turn half-right. My shadow cuts a hedge,
Climbs through a rhododendron to a porch,
And nods on a windowsill. How far it goes
I leave to burglars and Pythagoras.
Into the slanting glare I slant my watch,”
You’ll have to go over to Growney’s blog to read the whole thing.
I’ve been reading Growney’s math poetry blog for a while now, and I appreciate that as a mathematician, she has a deep understanding of what math is and isn’t. She doesn’t just blog about poems that contain numbers, and the poems she includes sometimes have subtle but deep mathematical connections. Her archives are certainly worth perusing. I particularly enjoyed the limericks about women mathematicians by Marion Deutsche Cohen and two posts from last winter about poetry made by rearranging words. Last Sunday’s post about how eating at McDonald’s is mathematically impossible is a reductio ad absurdum “proof” that effectively straddles a line between humor and seriousness.