# Calculus

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The population (in thousands) of the Tzitzit bird is well described by a function of the form P(t) = ae^kt, where t is the time in years and a and k are constants. If the population was 10 thousand when t-0 and 300 thousand when t=3, determine the constants a and k exactly. Then use the formula for P(t) to find the population when t=(4)

• Calculus -

10000 = ae^0 --> a=10000

300000 = 10000 e^3k
30 = e^3k
3k = ln30
k = 1/3 ln30

P(t) = 10000 * e^(t/3 ln30)
P(4) = 10000 * e^(4/3 ln30) = 932170

Note that since e^ln30 = 30,
P(t) = 10000 * 30^(t/3)
but to evaluate that, you need to fall back to logs anyway.

What it means is that the population grows by a factor of 30 every 3 years. But then, we knew that from the initial conditions given: P(3) = 30*P()

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