Question

8). a city is representing its projected growth rate by the equation given below where y represents the population and x represents the number of years since january 1, 2017

$y = 126,000(.94)^x$

circle all of the true statements about the population of the city:
a. there are 126,000 people that live in the city on january 1, 2017
b. there are 126,000 people that live in the city on january 1, 2018
c. the city is projected to decrease in population by 6% per year
d. the city is projected to increase in population by 94% per year

Explanation:

Step1: Analyze statement A

When \(x = 0\) (January 1, 2017), \(y=126000(0.94)^{0}\). Since \(a^{0}=1\) for \(a
eq0\), \(y = 126000\). So statement A is true.

Step2: Analyze statement B

When \(x = 1\) (January 1, 2018), \(y=126000(0.94)^{1}=126000\times0.94 = 118440
eq126000\). So statement B is false.

Step3: Analyze statements C and D

The general form of an exponential function is \(y = a(1 + r)^{x}\) (growth) or \(y=a(1 - r)^{x}\) (decay), where \(a\) is the initial amount and \(r\) is the rate. Here \(y = 126000(0.94)^{x}=126000(1 - 0.06)^{x}\). The rate of change \(r = 0.06=6\%\) (decay). So statement C is true and statement D is false.

Answer:

A. There are 126,000 people that live in the city on January 1, 2017; C. The city is projected to decrease in population by 6% per year.