in 2012, the population of city had been growing steadily, and the exponential function defined by ( f(t)=42.4cdot1.06^{t} ) models the population from 2005 to 2012, where t represents the number of years since 2005 and ( f(t) ) represents the population of the city, measured in thousands of people. use the given information to answer parts a through f.
d. the function is increasing because in the model, ( b 1 ).
b. determine the annual growth or decay rate from the model. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. it is a growth function with growth rate, ( r = 6% ).
(type an integer or a decimal.)
b. it is a decay function with decay rate, ( r=% ).
(type an integer or a decimal.)
c. according to the model, what is the initial value? what does the value mean in this situation? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the initial value is. this means that the population of the city in 2005 was approximately thousand people. (type an integer or a decimal.)
b. the initial value is. this means that the population of the city increased by approximately thousand people per year. (type an integer or a decimal.)

Question

in 2012, the population of city had been growing steadily, and the exponential function defined by ( f(t)=42.4cdot1.06^{t} ) models the population from 2005 to 2012, where t represents the number of years since 2005 and ( f(t) ) represents the population of the city, measured in thousands of people. use the given information to answer parts a through f.
d. the function is increasing because in the model, ( b > 1 ).
b. determine the annual growth or decay rate from the model. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. it is a growth function with growth rate, ( r = 6% ).
(type an integer or a decimal.)
b. it is a decay function with decay rate, ( r=% ).
(type an integer or a decimal.)
c. according to the model, what is the initial value? what does the value mean in this situation? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the initial value is. this means that the population of the city in 2005 was approximately thousand people. (type an integer or a decimal.)
b. the initial value is. this means that the population of the city increased by approximately thousand people per year. (type an integer or a decimal.)

Accepted Answer

# Answer:
A. The initial value is $42.4$. This means that the population of the city in 2005 was approximately $42.4$ thousand people.

# Explanation:
## Step1: Recall the form of an exponential function
The general form of an exponential function is $y = a\cdot b^{t}$, where $a$ is the initial value (when $t = 0$) and $b$ is the base.
## Step2: Identify the initial value
In the given function $f(t)=42.4\cdot1.06^{t}$, when $t = 0$ (the year 2005, since $t$ represents the number of years since 2005), we have $f(0)=42.4\cdot1.06^{0}$.
Since any non - zero number to the power of $0$ is $1$ ($a^{0}=1,a
eq0$), then $f(0)=42.4\times1 = 42.4$.
The coefficient $a = 42.4$ in the function $y=a\cdot b^{t}$ gives the value of the function at $t = 0$. In the context of the population model $f(t)$ (where $f(t)$ is the population in thousands of people), when $t = 0$ (year 2005), $f(0)$ represents the population of the city in 2005.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the form of an exponential function

Step2: Identify the initial value

Answer: