Chapter 6.6, Problem 36PPE

a.

To determine

To determine graph and expression of the inequalities that describes this situation.

Explanation of Solution

Given:

The price of gold is about $25 per gram

The price of silver is approximately $0.40 per gram

The total mass must be more than 10 g but less than 20g

The ring must contain at least 2 g of gold

The total cost of the gold and silver must be less than $90

Calculation for graph:

Let $y$ be the amount of silver in gram and $x$ be the amount of gold in gram.

The price of gold is about $25 per gram so cost is $ $25x$ .

The price of silver is approximately $0.40 per gram so cost ofis $0. $40y$ .

Total cost is $=25x+0.40y$

The total mass must be more than 10 g but less than 20g

So,

  $\Rightarrow 10<x+y<20$

The total cost of the gold and silver must be less than $90

Total cost is $=25x+0.40y$

So,

  $\Rightarrow 25x+0.40y<90$

Converting all the equation in slope interception form

INEQUALITY I: $y>-x+10$

The slope is -1 & the y -intercept is 10.

The shaded area is above the dashed boundary line.

INEQUALITY II: $y<-x+20$

The slope is -1 & the y -intercept is 20.

The shaded area is below the dashed boundary line.

INEQUALITY III: $y<-62.5x+225$

The slope is -62.5 & the y -intercept is 20.

The shaded area is below the dashed boundary line.

Graph:

  

Here, $x$ and $y$ are always positive $x\ge 0$ and $y\ge 0$ .

The solution is the intersection of the shaded regions in yellow and in black boundary.

Interpretation:

Therefore, the graph can be use to determine graph and expression of the inequalities that describes this situation.

b.

To determine

To determinethe mass of the ring and the cost of the gold and silver.

Answer to Problem 36PPE

Explanation of Solution

Given:

The price of gold is about $25 per gram

The price of silver is approximately $0.40 per gram

The total mass must be more than 10 g but less than 20g

The ring must contain at least 2 g of gold

The total cost of the gold and silver must be less than $90

Calculation for graph:

From the above expression,

  $\Rightarrow y>-x+10$

  $\Rightarrow y<-x+20$

  $\Rightarrow y<-62.5x+225$

  $\Rightarrow 22x+15y=100$

Let value of $x=2$ and $y=15$

  $\Rightarrow 10<x+y<20$

Values of x	Value of y	Mass
2	15	17

  $\Rightarrow 25x+0.40y<90$

Values of x	Value of y	Cost
2	15	56

Now plotting the graph,

Graph:

  

Here, $x$ and $y$ are always positive $x\ge 0$ and $y\ge 0$ .

The solution is the intersection of the shaded regions in yellow and in black boundary.

Interpretation:

Therefore,the mass of the ring and the cost of the gold and silver for 2gram of gold and 15gram of silver17gram and $56 respectively.