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?

Question

7411 views around the world

You can reuse this answer
Creative Commons License

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}#