In a particular region of a national park, there are currently 330 deer, and the population is increasing at an annual rate of 11%.
1. Write an exponential function to model the deer population in terms of the number of years from now.
2. Explain what each value in the model represents.
3. Predict the number of deer that will be in the region after five years. Show your work.
1. The exponential function to model the deer population in terms of the number of years from now would be:
P(t) = P₀ * (1 + r)^t
Where:
- P(t) represents the deer population after t years.
- P₀ represents the initial deer population (330 in this case).
- r represents the annual growth rate as a decimal (11% would be 0.11).
- t represents the number of years from now.
Therefore, the exponential function would be P(t) = 330 * (1 + 0.11)^t.
2. In the exponential function:
- P₀ (330) represents the initial population of deer in the region.
- r (0.11) represents the growth rate of the population per year, which is 11%.
- t represents the number of years from now, indicating how far into the future you want to predict the population.
3. To predict the number of deer that will be in the region after five years, substitute t = 5 into the exponential function:
P(5) = 330 * (1 + 0.11)^5
P(5) = 330 * (1.11)^5
P(5) = 330 * 1.735
P(5) = 571.05
Therefore, the predicted number of deer in the region after five years is approximately 571.