Car Talk Puzzler Solution: Man Growing A Lawn

I love car talk. My radio station in Miami plays reruns every Saturday (which I just learned were reruns, since I just learned they no longer produce new shows). Probably my favorite part of Car Talk is the puzzler, which is usually some math-based word problem or some word-based math problem. Last weekend’s puzzler was a great one. The short version goes like this:

“A man has a dirt yard. On June 1, he goes to a garden store and tries to figure out how he can get a nice, lush lawn in time for his July 4 party. After discussing his options, the store clerk suggests a new technique: a plug of grass that doubles every day. The clerk did the calculations and figured out that, if the man were to buy one plug, his lawn would be covered by June 30 (in 30 days). The man thinks that he would be cutting it too close, so he buys two plugs. How many days does it take for his yard to get covered?”

As a biologist/ecologist, I recognized this immediately. This is the problem of exponential population growth! You can write down a really simple equation. Suppose that today is day 1 (t1) and tomorrow is day 2 (t2). The population of tomorrow is twice that of today:

latex-image-1

The lawn size on day 3 (t3) is twice that of the lawn size on day 2:

latex-image-2

and so on and so on until you get to t(30). However, writing all of these calculations is time consuming and kind of a pain. So you can consolidate. If t(2) is twice that of t(1), and we know that t(1) is twice that of t(0), we can substitute in:

latex-image-3

which gives a general equation:

latex-image-4

where the population on the ith day is simply 2 raised to (i – 1) multiplied by the initial population size. So we can quickly calculate the lawn size on day 30:

latex-image-5

That’s the first bit of information we need. But the man started with two plugs (to be safe). We need to know how many days it takes to reach that same size:

latex-image-6

To solve for (i-1), we can use the awesomeness of (natural) logarithms:

latex-image-7

The Car Talk guys used nice numbers that led to nice integer solutions. The neat part is that, by starting with two plugs “for insurance” and spending twice as much, the man only saved himself a single day. The proof of that is in this plot:

figure_1

The two plug line (blue) reaches the final size of the one plug line (green) only one day before

This puzzler was easy for me because I recognized the word problem immediately as the exponential growth of populations, something I’m fairly familiar with.