Question

at the beginning of a population study, a city had 260,000 people. each year since, the population has grown by 9.8%. let t be the number of years since start of the study. let y be the citys population. write an exponential function showing the relationship between y and t.

Explanation:

Step1: Recall exponential growth formula

The general form of an exponential growth function is \( y = a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth rate (in decimal), and \( t \) is the time.

Step2: Identify values of \( a \) and \( r \)

Here, the initial population \( a = 260000 \). The growth rate \( r = 9.8\% = 0.098 \).

Step3: Substitute values into the formula

Substitute \( a = 260000 \) and \( r = 0.098 \) into \( y = a(1 + r)^t \), we get \( y = 260000(1 + 0.098)^t = 260000(1.098)^t \).

Answer:

\( y = 260000(1.098)^t \)