If working on a computer, open a new worksheet to model the situation.
Groups: Work in pairs on Problems B4-B6.
Think about a strategy for calculating how long the population takes to double. The Fill Down command on the spreadsheet can be used until a population of 700 is reached. To answer the question of whether it depends on the initial population, change that starting number in the spreadsheet and see if it doubles in the same place. The doubling time does not depend on the starting value; thus an exponent n can be found so that 1.14n = 2.
Here's one way to see that the time to double doesn't depend on the starting value, and it also highlights some important algebraic thinking.
You're looking for a year where 1.14 x 1.14 x 1.14x ... x 1.14 x n = 2 x n. There is an n multiplied on each side, so the only thing that could possibly make the multiple of 2 is all those 1.14s multiplied together. You just have to find the right number of them, and the number of 1.14s only depends on the year.
This also tells you that, for example, if you get a 5 percent raise at your job every year, the number of years it takes you to double your salary is fixed, and it doesn't depend on how much you start out earning.
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