System of Inequalities:
7x+11y ≤ 200
3x+4y ≤ 80
B
b
Graph:
C
c
Regular Doghouses: 0 Deluxe Doghouses: 18
Practice makes perfect
a
We will write a system of inequalities to define the number of products of a company for two types of doghouses: regular and deluxe. Let's first make a table to show the required hours to build and also to paint the corresponding doghouses.
Hours for Building
Hours for Painting
Regular Doghouse
7
3
Deluxe Doghouse
11
4
Let's say that the number of regular doghouses is x and the number of deluxe doghouses is y. Having the corresponding variables for the number of doghouses, we can write an expression to represent the total time to build and also to paint these houses.
Building: & 7 x + 11 y
Painting: & 3 x +4 y
Great! Next, we also know that the company employs 5 builders and 2 painters who can work a maximum 40 hours each. By knowing this, we can calculate the total hours for building and painting by multiplying the number of employees by 40. Let's do it!
Building: & 5 * 40 = 200
Painting: & 2 * 40 = 80
Note that we have a maximum of 200 hours to build and a maximum of 80 hours to paint both type of the houses. This means that the number of total hours will be less than or equal to 200 for building and less than or equal to 80 for painting. Now we can express this with a system of inequalities.
Building: & 7 x + 11 y ≤ 200
Painting: & 3 x +4 y ≤ 80
Now that we have our boundary line y= - 711x+ 20011, let's go ahead and graph it. To do so we will draw our line using our slope: a rise of 7 units in the negative direction and a run of 11 units in the positive direction. Moreover, since 20011 is approximately equal to 18.2, we will put our y-intercept near the point (0, 18).
Next, we need to decide which side of the function should be shaded. We can test this by substituting any point into the inequality and seeing if it holds true. Let's try with (0,0).
Notice that the point (0,0) is a solution to the inequality. Therefore, we should shade the side of the function containing that point.
Great! Now, we will apply the same process one more time for the second inequality of our system.
3 x + 4 y ≤ 80
Let's first write this inequality in the slope-intercept form.
Since (0,0) have satisfied our inequality, we will shade the side of the function containing that point.
Finally, we can shade the overlapping region of the individual solution sets to show the solution of the system.
c
We are told that a regular doghouse sells for $ 100 and a deluxe one sells for $ 200. We will find the corresponding number of houses to maximize the sales in one week. To do so we need to select the maximum number of deluxe houses because it makes more money. Let's examine the overlapping region that we found in Part B.
Notice that we are looking for the maximum number of deluxe houses and the minimum number of regular houses in the shaded area. Since we need to sell a whole number of houses in the context of the problem, we can select the point ( 0, 18).
Regular Houses: 0
Deluxe Houses: 18
To maximize the sales, 18 deluxe doghouses and 0 regular doghouse will be sold in one week.
