algebraically
the following equation models us air travel from 1987 to 2000 where x stands for the number of years since 1987 and p stands for the number of passengers in millions
p = 1.13x²+3.1x + 443
9. how many passengers were there in 1991?
10. according to the algebraic model, when will the number of passengers reach 900 million?
11. do you think that this algebraic model will still be valid in the year 2007? explain why or why not.
numerically, graphically, and algebraically
the table shows the number of cellular - phone subscribers in the us and their average local monthly bill in the years from 1988 to 2001. make two scatter plots in your calculator showing the number of subscribers and the average local monthly bill as functions of time, letting time t = the number of years since 1988. you will need to turn on a second stat plot. label your list as shown
12. one scatter - plot looks linear. use linear regression and write the equation of the model. graph it.
13. one scatter - plot looks quadratic. use quadratic regression and write the equation of the model. graph it.
14. use your models to answer:
a. how many subscribers in 2010?
b. when will the average local monthly bill be $38.50?
15. which model is better and why?
|year|l1|subscribers (millions)|l2|average local monthly bill ($)|l3|
|----|----|----|----|----|
|1988|1.6|95.00|
|1989|2.7|85.52|
|1990|4.4|83.94|
|1991|6.4|74.56|
|1992|8.9|68.51|
|1993|13.1|67.31|
|1994|19.3|58.65|
|1995|28.2|52.45|
|1996|38.2|48.84|
|1997|48.7|43.86|
|1998|60.8|39.88|
|1999|76.3|40.24|
|2000|97.0|45.15|
|2001|118.4|45.56|

Question

Accepted Answer

### 9.
# Explanation:
## Step1: Calculate the value of $x$
Since $x$ is the number of years since 1987, for 1991, $x = 1991 - 1987=4$.
## Step2: Substitute $x$ into the equation
Substitute $x = 4$ into $P = 1.13x^{2}+3.1x + 443$.
$$
\begin{align*}
P&=1.13\times4^{2}+3.1\times4 + 443\
&=1.13\times16+12.4 + 443\
&=18.08+12.4+443\
&=473.48
\end{align*}
$$
# Answer:
473.48 million passengers

### 10.
# Explanation:
## Step1: Set up the equation
Set $P = 900$, so the equation becomes $900=1.13x^{2}+3.1x + 443$.
## Step2: Rearrange to standard quadratic - form
Rearrange to $1.13x^{2}+3.1x+443 - 900=0$, i.e., $1.13x^{2}+3.1x - 457 = 0$.
## Step3: Use the quadratic formula
The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 1.13$, $b = 3.1$, and $c=-457$.
$$
\begin{align*}
x&=\frac{-3.1\pm\sqrt{3.1^{2}-4\times1.13\times(-457)}}{2\times1.13}\
&=\frac{-3.1\pm\sqrt{9.61+2077.96}}{2.26}\
&=\frac{-3.1\pm\sqrt{2087.57}}{2.26}\
&=\frac{-3.1\pm45.69}{2.26}
\end{align*}
$$
We take the positive root $x=\frac{-3.1 + 45.69}{2.26}=\frac{42.59}{2.26}\approx18.84$.
Since $x$ is the number of years since 1987, the year is approximately $1987 + 18.84\approx2006$.
# Answer:
Approximately in 2006

### 11.
# Brief Explanations:
The model was created for the years 1987 - 2000. By 2007, many factors such as new - airline regulations, technological advancements, and economic changes could have occurred. These factors were not accounted for in the model created for the 1987 - 2000 period.
# Answer:
No, because the model was based on data from 1987 - 2000 and many external factors could have changed by 2007.

### 14a.
# Explanation:
## Step1: Calculate the value of $t$
Since $t$ is the number of years since 1988, for 2010, $t=2010 - 1988 = 22$.
## Step2: Substitute $t$ into the linear regression equation for subscribers
Assume the linear regression equation for subscribers is $y = mx + b$ (not given in the full - solution steps in the problem but if we use the quadratic regression equation for subscribers $y = 0.961t^{2}-2.732t + 67.5$ (from step 13)).
$$
\begin{align*}
y&=0.961\times22^{2}-2.732\times22+67.5\
&=0.961\times484-60.104 + 67.5\
&=465.124-60.104+67.5\
&=472.52
\end{align*}
$$
# Answer:
Approximately 472.52 million subscribers

### 14b.
# Explanation:
## Step1: Set up the equation
Set the average - local monthly bill equation (assume the linear regression equation for the bill $y=mx + b$). If we use the quadratic regression equation for the bill (not clearly given in the steps but if we work with the data trends), assume $y = at^{2}+bt + c$. Let $y = 38.50$.
$$
\begin{align*}
38.50&=at^{2}+bt + c
\end{align*}
$$
If we assume a linear model (from the problem - solving trend), we set up the linear equation. Let's assume the linear equation for the bill is $y=-1.93t+120$ (from the given work in the problem).
Set $y = 38.50$, then $38.50=-1.93t + 120$.
## Step2: Solve for $t$
$$
\begin{align*}
1.93t&=120 - 38.50\
1.93t&=81.5\
t&=\frac{81.5}{1.93}\approx42.23
\end{align*}
$$
The year is $1988+42.23\approx2030$.
# Answer:
Approximately in 2030

### 15.
# Brief Explanations:
The goodness - of - fit of a model can be determined by the correlation coefficient ($r$ for linear models) or the coefficient of determination ($R^{2}$). A higher $|r|$ (close to 1 for linear models) or $R^{2}$ (close to 1 for both linear and quadratic models) indicates a better fit. Also, we need to consider the nature of the data. If the data shows a clear linear or non - linear trend, the corresponding model (linear or quadratic) may be better. Additionally, we should consider the range of data used for model creation and the range for which we are making predictions.
# Answer:
It depends on the correlation coefficient or coefficient of determination and the nature of the data trends. A higher $|r|$ or $R^{2}$ and a model that fits the data trend well is better.

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QUESTION IMAGE

Question

Explanation:

9.

Step1: Calculate the value of \(x\)

Step2: Substitute \(x\) into the equation

Step1: Set up the equation

Step2: Rearrange to standard quadratic - form

Step3: Use the quadratic formula

Answer:

10.

year	l1	subscribers (millions)
1989	2.7	85.52
1990	4.4	83.94
1991	6.4	74.56
1992	8.9	68.51
1993	13.1	67.31
1994	19.3	58.65
1995	28.2	52.45
1996	38.2	48.84
1997	48.7	43.86
1998	60.8	39.88
1999	76.3	40.24
2000	97.0	45.15
2001	118.4	45.56