Q. 52 - Precalculus Enhanced with Graphing Utilities

Question

Developing a Linear Model from Data The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with $60. 

Soda, s	Hot Dog, h
20	0
15	3
10	6
5	9

(a) Plot the ordered pairs $(s, h)$ in a Cartesian plane.

(b) Show that the number of hot dogs purchased h is a linear function of the number of sodas purchased s. 

(c) Determine the linear function that describes the relation between s and h. 

(d) What is the implied domain of the linear function? 

(e) Graph the linear function in the Cartesian plane drawn in part (a). 

(f) Interpret the slope. 

(g) Interpret the values of the intercepts. 

Step-by-Step Solution

Answer

Part (a) The graph is as follows,

Part (b) The function is linear. 

Part (c) The function is $h\left(s\right)=-0.6s+12$.

Part (d) The domain of the function is $\left\{s|0\le s\le 20\right\}$.

 

Part (e) The graph is 

 

Part (f) Number of sodas decrease by 0.6 per hot dog. 

Part (g)  

• This represent number of sodas are 0 when number of hot dogs are 12. 
• This represent number of sodas are 20 when number of hot dogs are 0. 

Part (a) Step 1. Given information

The data from table represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with $60. 

Soda, s	Hot Dog, h
20	0
15	3
10	6
5	9

Part (a) Step 2. Explanation.

The graph of ordered pairs $(s,h)$ in Cartesian plane.

Part (b) Step 1. Explanation.

Show that the number of hot dogs purchased h is a linear function of the number of sodas purchased s. 

Find average rate of change.

Consider two points $\left(20,0\right),\left(15,3\right)$

$$\begin{gathered} \frac{∆y}{∆x}=\frac{3-0}{15-20} \\ =-\frac{3}{5} \\ =-0.6 \end{gathered}$$

Consider two points $\left(15,3\right),\left(10,6\right)$

$$\begin{gathered} \frac{∆y}{∆x}=\frac{6-3}{10-15} \\ =-\frac{3}{5} \\ =-0.6 \end{gathered}$$

Consider two points $\left(10,6\right), \left(5,9\right)$

$$\begin{gathered} \frac{∆y}{∆x}=\frac{9-6}{5-10} \\ =-\frac{3}{5} \\ =-0.6 \end{gathered}$$

Here, $\frac{∆y}{∆x}$ is constant. So the function is linear. 

Part (c) Step 1. Explanation.

Write the linear function that describes the relation between s and h.  

The standard form of linear function is $y=mx+b$.Where m is slope and b is y-intercept.

We $\frac{∆y}{∆x}=-0.6$ This is slope of the function.

So the function becomes $h\left(s\right)=-0.6s+b$.

To find b, take any arbitrary point  $\left(20,0\right)$,

$$\begin{gathered} h\left(s\right)=-0.6\left(20\right)+b \\ 0=-12+b \\ 12=b \end{gathered}$$

The linear function that describes the relation between s and h is  $h\left(s\right)=-0.6s+12$.

Part (d) Step 1. Explanation.

Find the implied domain of the linear function.

The number of soda purchased can not be less than 0.

So $s\ge 0\Rightarrow h\left(s\right)\ge 0$

Hence,

$$\begin{gathered} h\left(s\right)\ge 0 \\ -0.6s+12\ge 0 \\ -0.6s\ge -12 \\ s\ge 20 \end{gathered}$$

The implied domain of the linear function is $\left\{s|0\le s\le 20\right\}$

 

Part (e) Step 1. Explanation.

The graph is as follows,

Part (f) Step 1. Explanation.

Interpret the slope. 

The slope of the function is -0.6.

As slope is negative. the function is decreasing. 

Here, Number of sodas decrease by 0.6 per hot dog.

Part (g) Step 1. Explanation.

Interpret the values of the intercepts. 

In the function $h\left(s\right)=-0.6s+12$, y-intercept is 12.

• This represent number of sodas are 0 then number of hot dogs are 12.

In the function $h\left(s\right)=0$ then $s=20$, This is x-intercept.

• This represent number of sodas are 20 when number of hot dogs are 0.