Question

write a function in terms of t that represents the situation. a population of 100,000 decreases by 2% each year. y = \square

Explanation:

Step1: Identify the formula for exponential decay

The general formula for exponential decay is \( y = a(1 - r)^t \), where \( a \) is the initial amount, \( r \) is the rate of decrease (in decimal form), and \( t \) is the time.

Step2: Determine the values of \( a \) and \( r \)

Here, the initial population \( a = 100000 \), and the rate of decrease \( r = 2\% = 0.02 \).

Step3: Substitute the values into the formula

Substituting \( a = 100000 \) and \( r = 0.02 \) into the exponential decay formula, we get \( y = 100000(1 - 0.02)^t \).
Simplifying \( 1 - 0.02 \) gives \( 0.98 \), so the function becomes \( y = 100000(0.98)^t \).

Answer:

\( y = 100000(0.98)^t \)