Question

guided practice
exponential models
in 1990, the population of washington d.c. was about 604,000 people. since then, the population has decreased about 1.8% each year. which exponential function best models this situation?
a. $y = 604000(0.982)^x$
b. $y = 604000(1.018)^x$
c. $y = (0.982)^x$
d. $y = (604000 \cdot 0.982)^x$

Explanation:

Step1: Recall exponential decay formula

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

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

Here, \( a = 604000 \) (initial population), \( r = 1.8\% = 0.018 \). So \( 1 - r = 1 - 0.018 = 0.982 \).

Step3: Form the equation

Substitute \( a = 604000 \) and \( 1 - r = 0.982 \) into the decay formula: \( y = 604000(0.982)^x \), which matches option A.

Answer:

A. \( y = 604000(0.982)^x \)