Answer:
Part e) [tex]501,170\ people[/tex]
Part f) The decay factor is 0.98
Part g) Decreasing
Part h) [tex]y=25,400(1+0.11)^x[/tex]
Part i) [tex]345,071\ people[/tex]
Step-by-step explanation:
we have
[tex]f(x)=501,170(0.98)^x[/tex]
where
x ----> is the number of years since 1995
f(x) ----> is the population of a Texas city
Part e) What was the population in 1995?
we know that
The equation of a exponential function decay is of the form
[tex]y=a(1-r)^x[/tex]
where
a represent the initial value (y-intercept of the function)
therefore
In the given function
The initial value is the value of the function when the value of x is equal to zero
so
For x=0
[tex]f(x)=501,170(0.98)^0=501,170\ people[/tex]
Part f) What is the growth/decay factor?
we know that
The equation of a exponential decay function is of the form
[tex]y=a(1-r)^x[/tex]
where
(1-r) ----> is the decay factor
In this problem we have
[tex]f(x)=501,170(0.98)^x[/tex]
(1-r)=0.98
therefore
The decay factor is 0.98
Part g) Is the population increasing or decreasing?
we know that
If the factor is greater than 1 then the population is increasing
If the factor is less than 1 and greater than zero, then the population is decreasing
In this problem
the factor is less than 1
0.98< 1 ----> is a decay factor
therefore
The population is decreasing
In the year 1995, the population of a town in Texas was recorded as 25,400 people. Each year since 1995, the population has increased on average by 11% each year
Part h) Write an exponential function to represent the town's population, y, based on the number of years that pass, x after 1995
we know that
The equation of a exponential growth function is of the form
[tex]y=a(1+r)^x[/tex]
we have
[tex]a=25,400\\r=11\%=11/100=0.11[/tex]
substitute
[tex]y=25,400(1+0.11)^x[/tex]
[tex]y=25,400(1.11)^x[/tex]
Part i) Based on the function, what is the population predicted to be in the year 2020?
Remember that the number of years is since 1995
so
x=2020-1995=25 years
substitute the value of x in the exponential function
[tex]y=25,400(1.11)^25=345,071\ people[/tex]
