Question

A forest has
\[800\] pine trees, but a disease is introduced that kills

\[\dfrac{1}{4}\] of the pine trees in the forest every year.
Write a function that gives the number of pine trees remaining
\[P(t)\] in the forest
\[t\] years after the disease is introduced.
\[P(t)=\]

Answer 1

To model the number of pine trees remaining in the forest after \( t \) years, given that \( \frac{1}{4} \) of the trees die each year, we can start with the initial number of trees and express the remaining trees after each year.

Initially, there are 800 pine trees. Each year, the forest loses \( \frac{1}{4} \) of the trees, leaving \( \frac{3}{4} \) of the trees remaining.

The number of trees after one year can be described as:
\[
P(1) = 800 \times \left(1 - \frac{1}{4}\right) = 800 \times \frac{3}{4} = 600
\]

After the second year, the number of trees remaining would be:
\[
P(2) = 600 \times \left(1 - \frac{1}{4}\right) = 600 \times \frac{3}{4} = 450
\]

Continuing this process, we can establish a general formula for the number of trees remaining after \( t \) years. After \( t \) years, the number of trees can be expressed as:
\[
P(t) = 800 \times \left(\frac{3}{4}\right)^t
\]

Therefore, the function that gives the number of pine trees remaining in the forest \( t \) years after the disease is introduced is:
\[
P(t) = 800 \times \left(\frac{3}{4}\right)^t
\]