Question

at the beginning of an environmental study, a forest covered an area of 1500 km². since then, this area has decreased by 5.25% each year. let t be the number of years since the start of the study. let y be the area that the forest covers in km². 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 decay (in decimal), and \( t \) is time.
Here, initial area \( a = 1500 \) km², decay rate \( r = 5.25\% = 0.0525 \).

Step2: Substitute Values into Formula

Substitute \( a = 1500 \) and \( r = 0.0525 \) into the formula:
\( y = 1500(1 - 0.0525)^t \)
Simplify \( 1 - 0.0525 = 0.9475 \), so the function is \( y = 1500(0.9475)^t \).

Answer:

\( y = 1500 \cdot (0.9475)^t \)