4.2 Modeling with linear functions  (Page 3/9)

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Using a linear model to investigate a town’s population

A town’s population has been growing linearly. In 2004, the population was 6,200. By 2009, the population had grown to 8,100. Assume this trend continues.

1. Predict the population in 2013.
2. Identify the year in which the population will reach 15,000.

The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y -intercept would correspond to the year 0, more than 2000 years ago!

To make computation a little nicer, we will define our input as the number of years since 2004.

To predict the population in 2013 ( $t=9$ ), we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.

To determine the rate of change, we will use the change in output per change in input.

The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to $\text{\hspace{0.17em}}t=0,$ giving the point $\text{\hspace{0.17em}}\left(0,\text{6200}\right).\text{\hspace{0.17em}}$ Notice that through our clever choice of variable definition, we have “given” ourselves the y -intercept of the function. The year 2009 would correspond to $\text{\hspace{0.17em}}t=\text{5,}$ giving the point $\text{\hspace{0.17em}}\left(5,\text{8100}\right).$

The two coordinate pairs are $\text{\hspace{0.17em}}\left(0,\text{6200}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,\text{8100}\right).\text{\hspace{0.17em}}$ Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.

We already know the y -intercept of the line, so we can immediately write the equation:

$P\left(t\right)=380t+6200$

To predict the population in 2013, we evaluate our function at $\text{\hspace{0.17em}}t=9.$

$\begin{array}{ccc}\hfill P\left(9\right)& =& 380\left(9\right)+6,200\hfill \\ & =& 9,620\hfill \end{array}$

If the trend continues, our model predicts a population of 9,620 in 2013.

To find when the population will reach 15,000, we can set $\text{\hspace{0.17em}}P\left(t\right)=15000\text{\hspace{0.17em}}$ and solve for $\text{\hspace{0.17em}}t.$

$\begin{array}{ccc}\hfill 15000& =& 380t+6200\hfill \\ \hfill 8800& =& 380t\hfill \\ \hfill t& \approx & 23.158\hfill \end{array}$

Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.

A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs$0.25 to produce each doughnut.

1. Write a linear model to represent the cost $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of the company as a function of $\text{\hspace{0.17em}}x,$ the number of doughnuts produced.
2. Find and interpret the y -intercept.

a. $\text{\hspace{0.17em}}C\left(x\right)=0.25x+25,000\text{\hspace{0.17em}}$ b. The y -intercept is $\text{\hspace{0.17em}}\left(0,25,000\right).\text{\hspace{0.17em}}$ If the company does not produce a single doughnut, they still incur a cost of \$25,000.

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.

1. Predict the population in 2014.
2. Identify the year in which the population will reach 54,000.
1. 41,100
2. 2020

Using a diagram to build a model

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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