In a particular region of a national park, there are currently 330 deer, and the population is increasing at an annual rate of 11%.
Write an exponential function to model the deer population.
Explain what each value in the model represents.
Predict the number of deer that will be in the region after five years. Show your work.
The exponential function to model the deer population is:
P(t) = 330(1 + 0.11)^t
Where:
P(t) = deer population after t years
330 = initial deer population
0.11 = annual growth rate (11% increase)
t = number of years
In this model, 330 represents the initial deer population in the region. The value of 0.11 represents the annual growth rate, indicating that the population is increasing by 11% each year. The variable t represents the number of years since the initial population count.
To predict the number of deer in the region after five years, we substitute t = 5 into the exponential function:
P(5) = 330(1 + 0.11)^5
P(5) = 330(1.11)^5
P(5) = 330(1.76234)
P(5) ≈ 582.735
Therefore, the predicted number of deer in the region after five years is approximately 583.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
A coordinate graph is shown. The horizontal axis extends from 0 to 12 years. The vertical axis extends from 0 to 9500 with an axis label of 'Value' in dollars. A curve is graphed which begins at left parenthesis 0 comma 3500 right parenthesis, then decreases passing through approximately left parenthesis 1 comma 2700 right parenthesis.
The exponential decay graph shows the expected depreciation for a new boat, selling for $3,500, over 10 years.
Write an exponential function for the graph. Use the function to find the value of the boat after 9.5 years.
To write an exponential function for the given graph, we can use the general form of an exponential decay function:
V(t) = V₀ * e^(-kt)
Where:
V(t) = value of the boat after t years
V₀ = initial value of the boat ($3,500)
e = Euler's number (approximately 2.71828)
k = decay rate
t = time in years
From the graph, the initial value of the boat is $3,500, so V₀ = 3500. The graph shows that after 1 year, the value of the boat decreases to approximately $2,700. This information allows us to find the decay rate k.
Using the given information for year 1:
V(1) = 3500 * e^(-k*1) = 2700
Solving for k:
3500e^(-k) = 2700
e^(-k) = 2700/3500
e^(-k) = 0.77142
-k = ln(0.77142)
-k ≈ -0.256
Substitute the decay rate k back into the exponential decay function:
V(t) = 3500 * e^(0.256t)
To find the value of the boat after 9.5 years, substitute t = 9.5 into the exponential decay function:
V(9.5) = 3500 * e^(0.256*9.5)
V(9.5) = 3500 * e^(2.4272)
V(9.5) ≈ 3500 * 11.326
V(9.5) ≈ 39641
Therefore, the value of the boat after 9.5 years is approximately $3,961.