- #1

jaypee

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The population of a midwestern city follows the exponential law. If the population decreased from 900,000 to 800,000 from 1993 to 1995, what will the population be in 1997?

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- Thread starter jaypee
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- #1

jaypee

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The population of a midwestern city follows the exponential law. If the population decreased from 900,000 to 800,000 from 1993 to 1995, what will the population be in 1997?

- #2

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600000 ?

- #3

- 128

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We assume that population change is proportional to the current population size. (This assumption holds until the population grows large enough that competition for resources occurs). We thus have a differential equations as follows:

dN(t)/dt = kN(t) , t is measured in years

with boundary conditions:

N(0) = 900,000 ; N(2) = 800,000

We can solve this by seperating the variables:

∫ dN(t)/N(t) = ∫ kdt

ln|N(t)| = kt +A , A is an arbitrary constant

N(t) = Be

Now we use the boundary conditions:

N(0) = A = 900,000

N(2) = 900,000e

=> k = -0.058891518

=> N(t) = 900,000e

We now use the equation to find the population at time t = 4 (1997):

N(4) ~ 711,111

Now, if you haven't done any work on differential equations, then all of the above may as well have been written in French. So I'll solve the problem using the info given:

You were told that the population grows exponentially, and the most general form for an exponential equations is:

N(t) = Ae

So we will use this equation and solve for A and k, as we did above:

N(0) = A = 900,000

N(2) = 900,000e

=> k = -0.058891518

=> N(t) = 900,000e

We now use the equation to find the population at time t = 4 (1997):

N(4) ~ 711,111

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