Question

in 2004, a forest covered an area of 1500 $\mathrm{km}^2$. since then, this area has decreased by 3.25\\% each year.
let $t$ be the number of years since 2004. 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 rate of decrease (in decimal), and \( t \) is the time.

Step2: Identify initial amount and rate

Here, the initial area \( a = 1500 \) \( \text{km}^2 \), and the rate of decrease \( r = 3.25\% = 0.0325 \).

Step3: Substitute values into the formula

Substitute \( a = 1500 \) and \( r = 0.0325 \) into the exponential decay formula: \( y = 1500(1 - 0.0325)^t \).
Simplify \( 1 - 0.0325 = 0.9675 \), so the function becomes \( y = 1500(0.9675)^t \).

Answer:

\( y = 1500(0.9675)^t \)