Chapter 3.3, Problem 4LC

Solution

To State:

The number of possible hours one can spend on each activity in one day in the given situation.

  $\begin{array}{ll}\text{Hours spent watching TV}\hfill & \text{Hours spent playing football}\hfill \\ 0\hfill & 1,2,3\hfill \\ 1\hfill & 1,2\hfill \\ 2\hfill & 1\hfill \end{array}$

The above table lists the number of hours spent in the given activities that satisfy the given conditions.

Given:

Spend no more than three hours in one day watching TV and playing football.

Play football for at least one hour each day.

Concepts Used:

Modelling a situational problem mathematically.

Finding whole number solutions for a set of linear inequalities.

Calculations:

Let number of hours watching TV be $x$ and the number of hours playing football be $y$ .

The total number of hours available for these activities is given to be $3$ . The total number of hours actually spent in these activities is $x+y$ . Represent this mathematically: $x+y\le 3$ .

The number of hours spent on playing football in a day must be at least $1$ . So $y$ must be at least $1$ . Represent this mathematically: $y\ge 1$ .

The situation is modelled by the inequalities:

  $\left\{\begin{array}{l}x+y\le 3\hfill \\ y\ge 1\hfill \end{array}\right.$

Make a table of whole number values of $x$ and $y$ that satisfy the first inequality.

  $\begin{array}{ll}x\hfill & y\hfill \\ 0\hfill & 0,1,2,3\hfill \\ 1\hfill & 0,1,2\hfill \\ 2\hfill & 0,1\hfill \\ 3\hfill & 0\hfill \end{array}$

Pick the values of $\left(x,y\right)$ pairs which satisfy $y\ge 1$ the second inequality.

  $\begin{array}{ll}x\hfill & y\hfill \\ 0\hfill & 1,2,3\hfill \\ 1\hfill & 1,2\hfill \\ 2\hfill & 1\hfill \end{array}$

The $\left(x,y\right)$ pairs in the above table satisfy the given conditions.

Conclusion:

  $\begin{array}{ll}\text{Hours spent watching TV}\hfill & \text{Hours spent playing football}\hfill \\ 0\hfill & 1,2,3\hfill \\ 1\hfill & 1,2\hfill \\ 2\hfill & 1\hfill \end{array}$

The above table lists the number of hours spent in the given activities that satisfy the given conditions.