Answer:
k ≈ 0.1733
Step-by-step explanation:
Using the numbers given, we find the multiplier of the population is 800/200 = 4 in 8 hours. This means the equation of growth can be written ...
y = 200(4^(t/8))
When this is written in the form ...
y = 200·e^(kt)
We can compare the two equations to see that ...
4^(1/8) = e^k
Taking natural logarithms, we have ...
1/8·ln(4) = k ≈ 0.1733
The growth rate k is approximately 0.1733.
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Additional comment
The comparison we were looking for was (4^(1/8))^t = (e^k)^t. Instead of "comparing" the equations, you could set them equal and solve for k.
200(4^(t/8)) = 200e^(kt)
4^(t/8) = e^(kt) . . . divide by 200
t/8·ln(4) = kt·ln(e) . . . . take natural logs
1/8·ln(4) = k . . . . . . use ln(e) = 1 and divide by t
