Question
There are
\[170\] deer on a reservation. The deer population is increasing at a rate of
\[30\%\] per year.
Write a function that gives the deer population
\[P(t)\] on the reservation
\[t\] years from now.
\[P(t)=\]
\[170\] deer on a reservation. The deer population is increasing at a rate of
\[30\%\] per year.
Write a function that gives the deer population
\[P(t)\] on the reservation
\[t\] years from now.
\[P(t)=\]
Answers
GPT-4o mini
To model the deer population \( P(t) \) on the reservation \( t \) years from now, we need to consider the initial population and the growth rate.
The initial deer population is given as \( P(0) = 170 \).
The deer population increases at a rate of \( 30\% \) per year. This means that each year, the population will be multiplied by \( 1 + 0.30 = 1.30 \).
The general formula for exponential growth can be expressed as:
\[
P(t) = P(0) \cdot (1 + r)^t
\]
where:
- \( P(0) \) is the initial population,
- \( r \) is the growth rate (as a decimal), and
- \( t \) is the time in years.
Substituting in the values:
\[
P(t) = 170 \cdot (1.30)^t
\]
Thus, the function that gives the deer population \( P(t) \) on the reservation \( t \) years from now is:
\[
P(t) = 170 \cdot (1.30)^t
\]
The initial deer population is given as \( P(0) = 170 \).
The deer population increases at a rate of \( 30\% \) per year. This means that each year, the population will be multiplied by \( 1 + 0.30 = 1.30 \).
The general formula for exponential growth can be expressed as:
\[
P(t) = P(0) \cdot (1 + r)^t
\]
where:
- \( P(0) \) is the initial population,
- \( r \) is the growth rate (as a decimal), and
- \( t \) is the time in years.
Substituting in the values:
\[
P(t) = 170 \cdot (1.30)^t
\]
Thus, the function that gives the deer population \( P(t) \) on the reservation \( t \) years from now is:
\[
P(t) = 170 \cdot (1.30)^t
\]