c Substitute the points from Part B into the revenue equation.
A
aSystem:0.5x+0.25y≤20 2x+3y≤120 x≥0 y≥0
Graph:
B
b (0,0),(40,0),(30,20),and(0,40)
C
cRevenues:
(0,0)→$0
(40,0)→$400
(30,20)→$460
(0,40)→$320 Maximum Revenue: $460 at (30,20)
Practice makes perfect
a We are told that we have dedicated 20 hours to crafting our goods for the craft fair. While we cannot go over that 20 hours, we can choose to quit early. We know that each necklace x takes 0.5 hours to make and each key chain y takes 0.25 hours to make. Let's write the total amount of time spent preparing for the fair.
0.5x+0.25y≤20
We are also told that we have saved $120 to buy materials for our goods. While we cannot go over that budget, we can choose to spend less. We know that each necklace x costs $2 to make and each key chain y costs $3 to make. Let's now write the total amount of money spent preparing for the fair.
2x+3y≤120
The exercise asks us to create a system with four inequalities, so we can write inequalities to represent the restrictions on the domain and range. We cannot make a negative number of necklaces or key chains. This gives us the following as our final system of inequalities.
0.5x+0.25y≤20 2x+3y≤120 x≥0 y≥0
We can graph this system by first creating boundary lines and then shading accordingly. Let's look at each inequality individually before we combine the graphs.
0.5x+0.25y≤20
Before we can graph the inequality we need to find the boundary line. To do this, we need to rewrite our inequality in slope-intercept form.
Now we can graph the boundary line, which will be solid because the symbol in our inequality is or equal to.
To choose on which side of the line we should shade, we can choose an arbitrary point and see if it satisfies the inequality. For simplicity's sake, let's choose (0,0).
Once again the boundary line will be solid because the symbol in our inequality is or equal to.
To choose on which side of the line we should shade, we can choose an arbitrary point and see if it satisfies the inequality. For simplicity's sake, let's choose (0,0).
Because (0,0) is a solution to the inequality, we should shade on the side of the line containing that point.
x≥0
Because this is just a vertical line, we can immediately graph the boundary line. We also know that we can shade to the right of the line, as it is simply values of x greater than or equal to 0.
y≥0
Because this is just a horizontal line, we can immediately graph the boundary line. We also know that we can shade above the line, as it is simply values of y greater than or equal to 0.
Combining the Inequalities
To find the final solution set for the system, we can graph all four inequalities on the same coordinate plane.
The overlapping area is the solution set. Let's look at that without any of the extra information.
b Using the graph created in Part A, we can label the corner points of the polygon made from the intersecting inequalities.
Let's write the vertices created by the system.
(0,0),(40,0),(30,20),and(0,40)
c We can find the revenue for each vertex by substituting the point into the given revenue equation. Let's do that and check which one produces the higher value of R.
(x,y)
10x+8y
R
( 0, 0)
10* 0+8* 0
0
( 40, 0)
10* 40+8* 0
400
( 30, 20)
10* 30+8* 20
460
( 0, 40)
10* 0+8* 40
320
The maximum revenue will be made if 30 necklaces and 20 key chains are made.
