Use the exponential growth model $P (t) = P_{0} e^{k t}$ $P(t) = P_0e^(kt)$ . How long will it take for the population of a certain country to double if its annual growth rate is 3.5%?

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Answer 1 · 2016-08-01T21:30:20

20.15 years to 2 dp

Let's assume the units of t are years. We want to find k.

$P (1) = 1.035 P_{0} = P_{0} e^{k}$ $P(1) = 1.035P_0 = P_0e^k$

$e^{k} = 1.035$ $e^k = 1.035$

${ln e}^{k} = ln (1.035)$ $lne^k = ln(1.035)$

$k ln (e) = ln (1.035)$ $kln(e) = ln(1.035)$

$\Rightarrow k = ln (1.035) y r^{- 1}$ $implies k = ln(1.035)yr^(-1)$

Now need $P (t) = 2 P_{0}$ $P(t) = 2P_0$

$2 P_{0} = P_{0} e^{k t}$ $2P_0 = P_0e^(kt)$

$e^{k t} = 2$ $e^(kt) = 2$

$k t = ln (2)$ $kt = ln(2)$

$t = \frac{ln (2)}{k} = \frac{ln (2)}{ln (1.035)}$ $t = ln(2)/k = ln(2)/ln(1.035)$

$t = 20.15$ $t = 20.15$ years

Use the exponential growth model P(t)=P0ektP(t) = P_0e^(kt). How long will it take for the population of a certain country to double if its annual growth rate is 3.5%?