# 4.2 Modeling with linear functions  (Page 8/9)

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If these trends were to continue, what would be the median home value in Mississippi in 2010?

$80,640 If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.) For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in [link] . Assume that the house values are changing linearly. Year Indiana Alabama 1950$37,700 $27,100 2000$94,300 $85,100 In which state have home values increased at a higher rate? Alabama If these trends were to continue, what would be the median home value in Indiana in 2010? If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.) 2328 ## Real-world applications In 2004, a school population was 1001. By 2008 the population had grown to 1697. Assume the population is changing linearly. 1. How much did the population grow between the year 2004 and 2008? 2. How long did it take the population to grow from 1001 students to 1697 students? 3. What is the average population growth per year? 4. What was the population in the year 2000? 5. Find an equation for the population, $\text{\hspace{0.17em}}P,$ of the school t years after 2000. 6. Using your equation, predict the population of the school in 2011. In 2003, a town’s population was 1431. By 2007 the population had grown to 2134. Assume the population is changing linearly. 1. How much did the population grow between the year 2003 and 2007? 2. How long did it take the population to grow from 1431 people to 2134 people? 3. What is the average population growth per year? 4. What was the population in the year 2000? 5. Find an equation for the population, $\text{\hspace{0.17em}}P,$ of the town $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years after 2000. 6. Using your equation, predict the population of the town in 2014. 1. $2134-1431=703\text{\hspace{0.17em}}$ people 2. $2007-2003=4\text{\hspace{0.17em}}$ years 3. Average rate of growth $\text{\hspace{0.17em}}=\frac{703}{4}=175.75\text{\hspace{0.17em}}$ people per year So, using $\text{\hspace{0.17em}}y=mx+b,$ we have $\text{\hspace{0.17em}}y=175.75x+1431.$ 4. The year 2000 corresponds to $\text{\hspace{0.17em}}t=-3.$ So, $\text{\hspace{0.17em}}y=175.75\left(-3\right)+1431=903.75\text{\hspace{0.17em}}$ or roughly 904 people in year 2000 5. If the year 2000 corresponds to $\text{\hspace{0.17em}}t\text{=0,}$ then we have ordered pair $\text{\hspace{0.17em}}\left(0,903.75\right)$ $y=175.75x+903.75\text{\hspace{0.17em}}$ corresponds to $\text{\hspace{0.17em}}P\left(t\right)=175.75t+903.75$ 6. The year 2014 corresponds to $\text{\hspace{0.17em}}t=14.\text{\hspace{0.17em}}$ Therefore, $\text{\hspace{0.17em}}P\left(14\right)=175.75\left(14\right)+903.75=3364.25$ . So, a population of 3364. A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be$71.50. If the customer uses 720 minutes, the monthly cost will be $118. 1. Find a linear equation for the monthly cost of the cell plan as a function of x , the number of monthly minutes used. 2. Interpret the slope and y -intercept of the equation. 3. Use your equation to find the total monthly cost if 687 minutes are used. A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of$10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be$17.80.

1. Find a linear equation for the monthly cost of the data plan as a function of $\text{\hspace{0.17em}}x,$ the number of MB used.
2. Interpret the slope and y -intercept of the equation.
3. Use your equation to find the total monthly cost if 250 MB are used.
1. 0.06 For every MB, the client is charged 6 cents. $\text{\hspace{0.17em}}\left(0,10\right)\text{\hspace{0.17em}}$ If no usage occurs, the client is charged \$10
2. $\begin{array}{ccc}\hfill C\left(250\right)& =& 0.06\left(250\right)+10\hfill \\ & =& 25\hfill \end{array}$

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