Question

E. What was the population in 1995? (the initial population for this situation) *

Answer 1

Answer:

Part E: C

Part F: A

Part G: B

Step-by-step explanation:

Answer 2

Answer:

Part e) $501,170\ people$

Part f) The decay factor is 0.98

Part g) Decreasing

Part h) $y=25,400(1+0.11)^x$

Part i) $345,071\ people$

Step-by-step explanation:

we have

$f(x)=501,170(0.98)^x$

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

$y=a(1-r)^x$

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

$f(x)=501,170(0.98)^0=501,170\ people$

Part f) What is the growth/decay factor?

we know that

The equation of a exponential decay function is of the form

$y=a(1-r)^x$

where

(1-r) ----> is the decay factor

In this problem we have

$f(x)=501,170(0.98)^x$

(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

$y=a(1+r)^x$

we have

$a=25,400\\r=11\%=11/100=0.11$

substitute

$y=25,400(1+0.11)^x$

$y=25,400(1.11)^x$

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

$y=25,400(1.11)^25=345,071\ people$