The bat population in a certain Midwestern county was 230,000 in 2009, and the observed doubling time for the population is 22 years. How do you find an exponential model?

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Answer 1 · 2017-01-12T15:24:32

#B = 230,000 * 2^(t/22)#

#B = 230,000 e^{ t/22 ln 2}#

let's say a quantity #Q#, starting with initial value #Q_o#, doubles over a period #tau#, we can say that:

#Q = Q_o * 2^(t/tau)#

so checking that:

So for the bats we can say that:

#B = 230,000 * 2^(t/22)#

If you want that in a calculus friendly form, you can instead start with the general idea of exponential growth/decay:

# (dQ)/(dt) = lambda Q#

#int_(Q_o)^Q (d Q)/Q = int_0^t lambda dt#

#Q = Q_o e^{lambda t}#

If we know that the amount doubles over a period #\tau# then:

#Q/ Q_o = 2 = e^{lambda tau}#

so #lambda = 1/tau ln 2#

And our relationship is:
#Q = Q_o e^{ t/tau ln 2}#

In that case, for the bats we say that:

#B = 230,000 e^{ t/22 ln 2}#