Question
A deer population in a forest starts with 100 and increases by 9% each year. Write an exponential function that models the data.
Answers
GPT-4o mini
To model the deer population with an exponential function, we can start with the general form of an exponential growth function:
\[ P(t) = P_0 \cdot (1 + r)^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal), and
- \( t \) is the time in years.
Given:
- The initial population \( P_0 = 100 \),
- The growth rate \( r = 9\% = 0.09 \).
Substituting these values into the formula, we get:
\[ P(t) = 100 \cdot (1 + 0.09)^t \]
This simplifies to:
\[ P(t) = 100 \cdot (1.09)^t \]
So the exponential function that models the deer population is:
\[ P(t) = 100 \cdot (1.09)^t \]
\[ P(t) = P_0 \cdot (1 + r)^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal), and
- \( t \) is the time in years.
Given:
- The initial population \( P_0 = 100 \),
- The growth rate \( r = 9\% = 0.09 \).
Substituting these values into the formula, we get:
\[ P(t) = 100 \cdot (1 + 0.09)^t \]
This simplifies to:
\[ P(t) = 100 \cdot (1.09)^t \]
So the exponential function that models the deer population is:
\[ P(t) = 100 \cdot (1.09)^t \]