Insta-Lawn Solution

Last week I posted a puzzler on the fictitious insta-lawn. The puzzler can be found at http://turf.uark.edu/turfhelp/archives/092909%20puzzler.html

The question was: If one plug would give him a lawn in 30 days, how quickly will two plugs do the job? XX people emailed in with the correct answer of: 1 less day (June 29).

If your first assumption was that the lawn would establish twice as fast (in only 15 days) then you were wrong. Here’s why.

All the extra plug does is gain him one day. Don’t forget, when you plant one plug, the next day, the amount of grass has already doubled in size, which is what you get at the end of the first day, if you had bought two of them.

Here’s the math if you still don’t believe me.

How much coverage (y) could be achieved from one plug doubling in size each day (x)?

y = 0.5e0.693x

When x = 30 days then y = 536,870,912
The number 536,870,912 is the number of times larger the plug is compared to its original size after 30 days)

In order to find out how many days it would take to fully establish the lawn if we buy two plugs, we need to first manipulate the original plugs size to twice it’s original value?

Step 1. Double to size (or number) of the original starting value to
y = 1.0e0.693x

Step 2. Now we have to solve for x when y = 536,870,912.
536,870,912 = 1.0e0.693x

Step 3. To solve for x we must take the natural log (ln) of both sides of the equation. This results in
20.101026824 = 0.69314718x

Step 4. Solve for x.
29 = x

So x = 29 days when we double the original number of plugs used to establish the lawn.

If you still don’t believe me, the exponential growth of the two scenarios is shown below.

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