Chapter 4.2, Problem 50HP

To determine

To Create:

A real-world situation that fits the graph shown.

Define the two quantities and describe the functional relationship between them.

Write an equation to represent this relationship and describe what the slope and y-intercept mean.

Answer to Problem 50HP

Equation of line is:

  $y=-\frac{1}{2}x+4$

Explanation of Solution

Given Graph:

  

Concept Used:

Equation of slope-intercept form of a line is given by:

  $\begin{array}{l}y=mx+b\mathrm{............................}(1)\\ where\\ b=y-intercept\\ m=\frac{raise}{run}=slope\end{array}$

Calculation:

REAL LIFE SITUATION DESCRIBING THE SITUATION IN GRAPH:

Ram drinks water from a bottle. Initially, the amount of water in the bottle was b liters.

If he drinks water at a rate of m liters per second. Then , we can form an equation depicting the situation in the graph and represent it in the form of slope-intercept form of linear equation.

Let

y represent the amount of water in the bottle .

x represent the time in seconds that water is being drunk out of the bottle

RELATION BETWEEN THE VARIABLES AND THE VALUES FROM THE GRAPH AS PER THE REAL-LIFE SITUATION:

Clearly, we can take points (0,4) , (2,3) from the graph .

(x,y) = (0,4) shows that :

Initially , when time is 0 seconds , the water in the bottle was 4 liters.

That is ,

y-intercept of the graph is b = 4.

Now , we find out the rate at which the water is being drunk from the bottle ( That is , Rate at which water level in the bottle is decreasing , initially which was 4 liters)

  $\begin{array}{l}y=mx+b\mathrm{...............................}[slope-intercept]\\ y=mx+\mathrm{4...............................}[b=4]\\ 3=m(2)+\mathrm{4............................}[(x,y)=(2,3)lie\_on\_line]\\ 3-4=2m+4-\mathrm{4....................}[subtract"4"each\_side]\\ -1=2m\mathrm{.................................}[simplify]\\ -\frac{1}{2}=\frac{2m}{2}\mathrm{................................}[divide"2"each\_side]\\ -\frac{1}{2}=m\\ or\\ m=-\frac{1}{2}\end{array}$

We can also interpret from the graph that from (0,4), we go down 1 unit and then go right 2 units , then we reach at point (2,3)

So, $slope=\frac{rise}{run}=-\frac{1}{2}$

  

WRITING THE EQUATION OF LINE :

  $\begin{array}{l}y=mx+b\mathrm{................................}[slope-\mathrm{int}ercept]\\ y=-\frac{1}{2}x+\mathrm{4.............................}[m=-\frac{1}{2},b=4]\end{array}$

So, the required equation is given by:

  $\begin{array}{l}y=-\frac{1}{2}x+4\\ \end{array}$

y represent the amount of water in the bottle

x represent the time in seconds that water is being drunk out of the bottle

THE INTERPRETATION OF SLOPE AND y-INTERCEPT:

The slope is the rate at which water is leaving the bottle .

That is , $\frac{1}{2}$ liters per second.

The y − intercept represents the amount of water initially in the bottle ( at 0 second) .

That is , 4 liters , when the bottle was full.