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Elephant Population Estimates—Namibia

Combined estimates for Etosha National Park and the Northwestern Population




Year		Base YearEstimated Number of Elephants

1998		33,218

2000		53,628

2002		73,721

2004		93,571






The elephant population in northwestern Namibia and Etosha National Park can be predicted by the expression $2,649(1.045)^b$, where $b$ is the number of years since 1995.



What does the value 2,649 represent?


AThe predicted increase in the number of elephants in the region each year.
BThe percentage the elephant population is predicted to increase each year.
CThe predicted number of elephants in the region in 1995.
DThe year when the elephant population is predicted to stop increasing.

Verified Answer :

  • The model \(P(b) = 2,649(1.045)^b\) is an exponential growth function where \(P_0 = 2,649\) represents the initial value.
  • Since \(b\) is the number of years since 1995, setting \(b=0\) gives the population in 1995: \(P(0) = 2,649(1.045)^0 = 2,649\).
  • This calculation identifies the constant 2,649 as the predicted population size in the base year of 1995.
  • Final answer: \(\boxed{\text{The predicted number of elephants in the region in 1995}}\).

Explanation

  1. Analyzing the Population Model

The provided mathematical model for the elephant population is given by the expression \(P(b) = 2,649(1.045)^b\), where \(P(b)\) represents the number of elephants and \(b\) represents the number of years since 1995. This is an example of an exponential growth model, which typically takes the general form \(P(b) = P_0(r)^b\), where \(P_0\) is the initial value at time \(b=0\), and \(r\) is the growth factor.

  1. Evaluating the Model at the Base Year

In the expression \(P(b) = 2,649(1.045)^b\), the variable \(b\) represents the number of years since 1995. By definition, at the starting year of 1995, the value of \(b\) is 0. To determine the meaning of the constant 2,649, we can evaluate the population function at \(b = 0\):
\[P(0) = 2,649 \times (1.045)^0 = 2,649 \times 1 = 2,649.\]
Since \(P(0)\) is the population when \(b=0\), and \(b=0\) corresponds to the base year 1995, it follows that 2,649 is the predicted number of elephants in the region in 1995.

  1. Conclusion

Comparing this result to the given options, we can conclude that the value 2,649 represents the predicted number of elephants in the region in 1995. The growth factor is 1.045, which implies an annual growth rate of 4.5%, but the constant term itself simply denotes the initial population count at the start of our observation period. Therefore, the correct choice is: The predicted number of elephants in the region in 1995.

Examples

Exponential models like the one used for the elephant population are fundamental tools in fields like biology, finance, and economics. For example, if a small startup company has an initial valuation of \(\$1,000,000\) and is expected to grow its revenue at a rate of 10% per year, its value after \(t\) years can be modeled as \(V(t) = 1,000,000(1.10)^t\). Understanding that the initial constant represents the starting point at \(t=0\) allows business analysts and scientists to make accurate predictions about future trends and understand the current state of a system relative to its beginning.

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