Question

in 2005, a forest covered an area of 1500 \\(\mathrm{km}^2\\). since then, this area has decreased by 8.25% each year. let \\(t\\) be the number of years since 2005. let \\(y\\) be the area that the forest covers in \\(\mathrm{km}^2\\). write an exponential function showing the relationship between \\(y\\) and \\(t\\).

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential decay function is $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the decay rate, and $t$ is time.

Step2: Identify given values

Initial area $a = 1500$ km², decay rate $r = 8.25\% = 0.0825$.

Step3: Substitute values into formula

Calculate $1 - r = 1 - 0.0825 = 0.9175$, then substitute into the formula.
$y = 1500(0.9175)^t$

Answer:

$y = 1500(0.9175)^t$