Chapter 7.5, Problem 26E

a.

To determine

To write: and graph a system of inequalities that describe the exercise target heart rate region.

Answer to Problem 26E

The graph will be

  

Explanation of Solution

Given:

The formula for a person’s maximum heart rate is $220-x$ , where x is the person’s age in years for $20\le x\le 70$ . It is given that when a person exercises, the person should strive for a heart rate that is at least 50% of the maximum and at most 85% of the maximum.

Concept Used:

Systems of inequalities that describe the exercise target heart rate region are graphed using the inequality equations.

Calculation:

Let, x is the age of the person and y is the heart rate of them.

The inequality equations as per the given information are as follows,

Person’s age $20\le x\le 70$

Minimum heart rate at 50% $y\ge 0.5\left(220-x\right)$

Maximum heart rate at 50% $y\le 0.85\left(220-x\right)$

These are the system of inequalities and their graph is as follows.

For the graph of $20\le x\le 70$

The region lies between the lines $x=20$ and $x=70$

For the graph of $y\ge 0.5\left(220-x\right)$ take $y=0.5\left(220-x\right)$

Make a table

  $\begin{array}{cccc}x& 220& 0& 100\\ y& 0& 110& 60\end{array}$

Plot the points and join them

  

For $y\ge 0.5\left(220-x\right)$ the region will be above the line $y=0.5\left(220-x\right)$ because the point (0,0) is not included in the region.

For the graph of $y\le 0.85\left(220-x\right)$ take $y=0.85\left(220-x\right)$

Make a table

  $\begin{array}{cccc}x& 220& 0& 100\\ y& 0& 187& 102\end{array}$

Plot the points and join them

  

For $y\le 0.85\left(220-x\right)$ the region will be below the line $y=0.85\left(220-x\right)$ because the point (0,0) is included in the region.

On combining all the facts above the final graph looks like

  

b.

To determine

To Find: two solutions of the system and interpret their meanings in the context of the problem.

Answer to Problem 26E

Two solutions of the system will be $\left(20,100\right)\text{and}\left(20,170\right)$

Explanation of Solution

Given:

The formula for a person’s maximum heart rate is $220-x$ , where x is the person’s age in years for $20\le x\le 70$ . It is given that when a person exercises, the person should strive for a heart rate that is at least 50% of the maximum and at most 85% of the maximum.

Also from the previous part the solution region is

  

Concept used:

Any two points in the solution region is two solutions of the system

Calculation:

From the above graph it can be observed that $\left(20,100\right),\left(70,75\right),\left(70,127.5\right)\text{and}\left(20,170\right)$ are the boundary points of the solution region and hence satisfies the solution

For $\left(20,100\right)$ , age of the person will be 20 years and heart rate of the person will be 100

Then the maximum heart rate of the same person will be $220-100=120$

Hence the person’s heart rate is according to the required criteria

For $\left(20,170\right)$ , age of the person will be 20 years and heart rate of the person will be 100

Then the maximum heart rate of the same person will be $220-170=50$

Hence the person’s heart rate is according to the required criteria