Chapter 6.6, Problem 3GP

a.

To determine

To find:Variables and inequalities of given situation

Answer to Problem 3GP

Variables are ‘ $s$ ’ be the number of Sweatshirts.

‘ $t$ ’ be the number of T-shirts

Inequalities are

  $\begin{array}{l}s+t\le 120\\ 22s+15t\ge 2000\end{array}$

Explanation of Solution

Given information:

Cost of each sweatshirt = $22

Cost of each T-shirt = $15

Maximum number of shirts that they can print = 120

Minimum Amount they need to earn in sales = $2000.

Let ‘ $s$ ’ be the number of Sweatshirts.

Let ‘ $t$ ’ be the number of T-shirts.

From given information

Maximum number of shirts that they can print = 120

Therefore

Number of sweatshirts + Number of T-shirts $\le 120$

  $\Rightarrow s+t\le 120$ --------(1)

From the given information

Minimum amount they need to earn in sales = $2000

Amount from sweatshirts + Amount from T-shirts $\ge 2000$

  $\Rightarrow 22\left(s\right)+15\left(t\right)\ge 2000$

  $\Rightarrow 22s+15t\ge 2000$ ---------(2)

These equation-(1) and equation-(2) represents the system of inequalities for the given situation.

b.

To determine

To find:Graph of the inequalities of Given situation.

Explanation of Solution

Given information:

Cost of each sweatshirt = $22

Cost of each T-shirt = $15

Maximum number of shirts that they can print = 120

Minimum Amount they need to earn in sales = $2000.

Inequalities are

  $\begin{array}{l}s+t\le 120\\ 22s+15t\ge 2000\end{array}$

Graph of given inequalities is

  

Here Blue colour represents the region of inequality $22s+15t\ge 2000$ .

Green colour represents the region of inequality $s+t\le 120$ .

Solution is the common region of Blue and Green colour.

c.

To determine

To find:One of the solution.

Answer to Problem 3GP

  $\left(100,10\right)$ .

Explanation of Solution

Given information:

Cost of each sweatshirt = $22

Cost of each T-shirt = $15

Maximum number of shirts that they can print = 120

Minimum Amount they need to earn in sales = $2000.

Graph of given inequalities is

  

Here Blue colour represents the region of inequality $22s+15t\ge 2000$ .

Green colour represents the region of inequality $s+t\le 120$ .

Solution is the common region of Blue and Green colour.

As $\left(100,10\right)$ is present in common region it is one of the solution of above inequalities.

d.

To determine

To find:Is $\left(45,30\right)$ a solution.

Answer to Problem 3GP

No.

Explanation of Solution

Given information:

Cost of each sweatshirt = $22

Cost of each T-shirt = $15

Maximum number of shirts that they can print = 120

Minimum Amount they need to earn in sales = $2000.

Graph of given inequalities is

  

Here Blue colour represents the region of inequality $22s+15t\ge 2000$ .

Green colour represents the region of inequality $s+t\le 120$ .

Solution is the common region of Blue and Green colour.

As $\left(45,30\right)$ is Not present in common region of both inequalities.

So it is not a solution.