Exercise 46, Page 240
Ozark Furniture Company can obtain at most 3000 board feet of maple lumber for making its classic and modern maple rocking chairs. A classic maple rocker requires 15 board feet of maple, and a modern rocker requires 12 board feet of maple. Write an inequality that limits the possible number of maple rockers of each type that can be made, and graph the inequality in the first quadrant.
Solution:
The most maple lumber that Ozark Furniture Company can obtain is 3000 board feet. The company is making two types of maple rocking chairs – classic and modern maple rocking chairs. To begin with, a variable needs to be assigned to each type of rocking chairs. For the classing maple rocking chair, the variable c is assigned, and for the modern maple rocking chair the variable m is assigned. What needs to be done next is to set an inequality that describes the different possibilities that the furniture company can have with the 3000 board feet of maple lumber. The company needs 15 board feet of maple to make one classic maple rocker. Moreover, there are c units of classic maple rockers, which are not known yet. Therefore, there will be 15c rockers, so that if the company decides to make one classic maple rocker, it will need 15 board feet of maple lumber. For the modern maple rockers, the company uses 12 board feet of maple lumber, which will give 12m rockers. In order to construct the inequality, 15c and 12m need to be added together, because Ozark Furniture Company wants to use the 3000 board feet of maple lumber for making both classic rockers and modern rockers. The total of the two types of rockers has to be less than or equal to 3000 board feet of maple lumber, because this is the whole amount that the company has. The inequality is represented in the following way:
15c+12m≤3000
15c+120≤3000
15c≤3000
c≤200
The maximum amount of classic maple rockers that the company can produce, given the total board feet of maple lumber, is equal to 200, which will be the intercept on the x-axis. The intercept for the modern maple rockers is fount the same way – by investigating how many chairs the company will be able to produce, if it decides to produce only modern maple rockers, which will make c equal to zero.
15(0)+12m≤3000
12m≤3000
m≤250
Therefore, the maximum amount of modern maple rocking chairs is equal to 250, which is also the y-intercept on the graph. Now that both the x- and the y-intercepts are present, the line 15c+12m=3000 can be constructed, which will be a straight down-sloping line. Since the inequality is 15c+12m≤3000, this is going to be represented by the shaded area under the just constructed line; between (0, x, y) or (0, 200, 250) in this case. The area describes all the different combinations of classic and modern maple rocking chairs that the company can make with a total of 3000 board feet of maple lumber.
When the point (100, 100) is investigated on the graph, it results in the following:
15*100+12*100=2700
2700≤3000
If the company decides to make 100 classic rocking chairs and 100 modern rocking chairs, it will use 2700 board feet of maple lumber, and it will have with 300 board feet left. This point will be inside the shaded area since 2700 is less than 3000 board feet.
When the point (150, 100) is investigated on the graph, it results in the following:
15*150+12*100=3450
3450≥3000
If the company decides to make 150 classic rocking chairs and 100 modern rocking chairs, it will use 3450 board feet of maple lumber, which means that it will need 450 board feet more of maple lumber. Therefore, this point is outside the shaded area and is not reachable for the company.
When the point (140, 75) is investigated on the graph, it results in the following:
15*140+12*75=3000
If Ozark Furniture Company decides to build 140 classic rocking chairs and 75 modern rocking chairs, it will use exactly 3000 board feet of maple lumber. Moreover, the company will not be able to make any mistakes since it will need all the maple lumber it has. This point is exactly on the line with coordinates (140, 75).
A chain furniture store faxes an order for 175 modern rocking chairs and 125 classic rocking chairs. In order to see if Ozark Furniture Company will fulfil this order, the numbers are plugged in the inequality:
15*125+12*175=3975
3975≥3000
Ozark Furniture Company will need 975 board feet more of maple lumber to fulfil this order of 175 modern rocking chairs and 125 classic rocking chairs. Therefore, it will not fulfil the order of the chain furniture store.
References:
Dugopolski, M. (2012) Elementary and Intermediate Algebra (4th Ed.). New York, NY: MacGraw-Hill Publishing
