Question

A forest has 800800800 pine trees, but a disease is introduced that kills \dfrac{1}{4} 4 1 ​ start fraction, 1, divided by, 4, end fraction of the pine trees in the forest every year. Write a function that gives the number of pine trees remaining P(t)P(t)P, left parenthesis, t, right parenthesis in the forest ttt years after the disease is introduced.

Answer 1

This is a perfect example of exponential decay. In this case the growth factor should be represented by a fraction, and it is! This forest, starting out with apparently ( 800? ) pine trees, has a disease spreading, which kills 1 / 4th of each of the pine trees yearly. Therefore, the pine trees remaining should be 3 / 4.

Respectively 3 / 4 should be the growth factor, starting with 800 pine trees - the start value. This can be represented as such,

$P( t ) = a( b )^t$ - where a = start value, b = growth factor, t = time ( variable quantity )

____

Thus, the function $P( t ) = 800( \frac{3}{4} )^t$ can model this problem. The forest after t years should have P( t ) number of pine trees remaining.