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The yearly growth rate of the US is 1 percent. Assuming this growth rate to be constant, in how many years will the population double?

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- #1

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The yearly growth rate of the US is 1 percent. Assuming this growth rate to be constant, in how many years will the population double?

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chroot

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[tex]N = N_0 e^{kt}[/tex]

Where [itex]N[/itex] is the population at time t, [itex]N_0[/itex] is the population a time 0, [itex]k[/itex] is the "growth constant," and [itex]t[/itex] is time.

The first step in getting a handle on these kinds of problems is finding k. Once you find k, you can easily find the population at any time t.

To find k, use the condition you already know: in one year, the population grows by one percent. That is, when t = 1, [itex]N = 1.01 N_0[/itex].

The equation is thus:

[tex]1.01 N_0 = N_0 e^{k \cdot 1}[/tex]

From which you should be able to readily calculate k.

To determine when the population doubles, use your known value of k and set the equation up in the following way:

[tex]2 N_0 = N_0 e^{kt}[/tex]

And solve for t.

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- #3

HallsofIvy

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Taking logs of both sides, log(2)=tlog(1.01) or t=log(2)/log(1.01).

In Chroot's method, k= ln(1.01) so this is, in fact, exactly the same answer. (I used log instead of ln because any base will do.)

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