What Will the Population Be in 1997 Based on Exponential Growth?

In summary, the population of the midwestern city in 1997 is estimated to be around 711,111 based on an exponential model and given data from 1993 and 1995. The model assumes that population change is proportional to the current population size, and the solution was found through differential equations. However, if you are not familiar with differential equations, the same result can be obtained by using the general form of an exponential equation and solving for the constants.
  • #1
jaypee
The following question is from a textbook..and I can't seem to solve it. Can someone help.
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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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  • #2
600000 ?
 
  • #3
I'll solve this problem from scratch, so that you can see how the mathematical model is formulated:

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) = Bekt

Now we use the boundary conditions:

N(0) = A = 900,000
N(2) = 900,000e2k = 800,000
=> k = -0.058891518
=> N(t) = 900,000e-0.058891518t

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) = Aekt , A and k are arbitrary constants

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

N(0) = A = 900,000
N(2) = 900,000e2k = 800,000
=> k = -0.058891518
=> N(t) = 900,000e-0.058891518t

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

N(4) ~ 711,111
 

Related to What Will the Population Be in 1997 Based on Exponential Growth?

What is population growth?

Population growth is the change in the number of individuals in a population over time. It can be positive, when there is an increase in the number of individuals, or negative, when there is a decrease in the number of individuals.

What factors influence population growth?

There are several factors that can influence population growth, such as birth rate, death rate, immigration, and emigration. Birth rate refers to the number of births in a population, while death rate refers to the number of deaths. Immigration is the movement of individuals into a population, while emigration is the movement of individuals out of a population.

How is population growth measured?

Population growth is typically measured using the crude birth rate, crude death rate, and net migration rate. These rates are calculated by dividing the number of births, deaths, and net migration by the total population.

What are the consequences of rapid population growth?

Rapid population growth can have several consequences, such as increased competition for resources, strain on infrastructure and public services, and environmental degradation. This can also lead to social and economic issues, such as poverty and inequality.

What are some strategies to manage population growth?

Some strategies to manage population growth include family planning, education and empowerment of women, promoting sustainable resource use, and implementing policies to address economic and social issues. These strategies can help to slow down population growth and promote sustainable development.

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