Stephanie M. answered 04/21/15
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We can find A and k by plugging in values. Start with the data from 1950, plugging 0 in for t and 2.555 in for P(t):
2.555 = Aek(0)
2.555 = Ae0
2.555 = A(1)
2.555 = A
Now that we know A, we can plug in the data from 2013, t = 63 and P(t) = 7.2, to find k:
7.2 = 2.555ek(63)
2.818 = e63k
ln(2.818) = 63k
1.036 = 63k
0.01644 = k (approximately)
The final equation is P(t) = 2.555e0.01644t
To find out how long it will take for the current population to triple, we need to know what x is when 2.555e0.01644(t+x) = 3(2.555e0.01644t). Let's solve:
2.555e0.01644(t+x) = 3(2.555)e0.01644t
e0.01644(t+x) = 3e0.01644t
(e0.01644(t+x)) / (e0.01644t) = 3
e0.01644(t+x) - 0.01644t = 3
e0.01644t + 0.01644x - 0.01644t = 3
e0.01644x = 3
ln3 = 0.01644x
1.0986 = 0.01644x
66.826 = x
So, the tripling time is 66.826 years.