Chapter 6.6, Problem 31PPE

To determine

To determine all the combination of T-shirt and dress shirt one can buy.

Explanation of Solution

Given:

$100 gift certificate

T-shirts for $15 and dress shirts for $22

Spend no more than the amount of the gift certificate

Want to leave at most $10 of the gift certificate unspent

Need at least one dress shirt

Calculation for graph:

Let $y$ be the number of T-shirts and $x$ be the number of dress shirts.

Each T-shirt cost $15 so cost of T-shirts is $ $15y$.

Each dress shirts cost $22 so cost of dress shirts is $ $22x$.

Total cost is $=22x+15y$

Want to spend no more than $100,

So,

  $\Rightarrow 22x+15y\le 100$

So, the boundary line expression is,

  $\Rightarrow 22x+15y=100$

Substituting $x=0$ the boundary line intercept is solving for $y$ ,

  $\Rightarrow 22(0)+15y=100$

Values of y	Value of x
6.7	0

Substituting $y=0$ the boundary line intercept is solving for $x$ ,

  $\Rightarrow 22x+15(0)=100$

Values of x	Value of y
4.5	0

Now plotting the graph,

Graph I:

  

Solving the inequality for $y$ ,

  $\Rightarrow 22x+15y\le 100$

  $\Rightarrow y\le \frac{100-22x}{15}$

Since, the expression is having $\le$ symbol so shaded region will be below the boundary line

Graph II:

  

Want to leave at most $10 of the gift certificate unspent which means $\$(100-10)=\$90$ ,

Therefore, expression is,

  $\Rightarrow 22x+15y\ge 90$

So, the boundary line expression is,

  $\Rightarrow 22x+15y=90$

Substituting $x=0$ the boundary line intercept is solving for $y$ ,

  $\Rightarrow 22(0)+15y=90$

Values of y	Value of x
6	0

Substituting $y=0$ the boundary line intercept is solving for $x$ ,

  $\Rightarrow 22x+15(0)=90$

Values of x	Value of y
4.1	0

Now plotting the graph,

Graph III:

  

Solving the inequality for $y$ ,

  $\Rightarrow 22x+15y\ge 90$

  $\Rightarrow y\ge \frac{90-22x}{15}$

Since, the expression is having $\ge$ symbol so shaded region will be below the boundary line

  

Need at least one dress shirt that is,

  $\Rightarrow x\ge 1$

The boundary line of this inequality is $x=1$ so it passes through $1$ in $x$ axis,

So the final graphical solution of the inequality is,

Graph IV:

  

So, the deep purple area with black border represents the all possible combination of T-shirt and dress shirt one can buy.

Interpretation:

Therefore, deep purple area with black border represents the all possible combination of T-shirt and dress shirt one can buy.