SOLUTION: The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. I

Question 266547: The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. If the current population is 15,000, in how many years is the population expected to be 37,500? Round to the nearest year
51
8
5
3

Found 2 solutions by drk, stanbon:
Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
By saying current population, we create the coordinate: (0, 15000). We put that into
(i)
to get
(ii)
which is simply
(iii)
so k = 14999
Now, we rewrite the equation with our new k to get
(iv)
We are given 37000 as our new population number, place that into the equation and solve for t. we get
(v)
subtract 1 and then divide by 14999 to get
(vi)
take an "LN" of both sides to get
(vii)
divide to get
(viii) years
-----
to the nearest year, it is 8.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke^(0.12t) where k is a constant and t is the time in years. If the current population is 15,000, in how many years is the population expected to be 37,500? Round to the nearest year
---
Use current population to find "k":
15000 = 1 +k*e^(0.12*0)
15000 = 1 + k
k = 14999
----------------------------------------
Equation:
P(t) = 14999e^(0.12t)
----
in how many years is the population expected to be 37,500?
37,000 = 14999*e^(0.12t)
2.467 = e^(0.12t)
Take the natural log of both sides to get:
0.12t = ln(2.467)
t = 7.52 years
Rounded up t = 8 years.
============================
Cheers,
Stan H.
51
8
5
3

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