Question Video: Identifying Regions Representing Inequalities Mathematics

A toy factory produces two types of planes, 2-engine planes and 4-engine planes. Each 2-engine plane requires 6 hours in the assembly department and one hour in the quality control department, and each 4-engine plane requires 8 hours in the assembly department and 2 hours in the quality control department.The maximum number of working hours per week is 120 in the assembly department and 25 in the quality control department. Which of the following is the graph representing the number of planes produced per month? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

A toy factory produces two types of planes, two-engine planes and four-engine planes. Each two-engine plane requires six hours in the assembly department and one hour in the quality control department. And each four-engine plane requires eight hours in the assembly department and two hours in the quality control department. The maximum number of working hours per week is 120 in the assembly department and 25 in the quality control department. Which of the following is the graph representing the number of planes produced per week?

And then we have five quite small graphs. We’ll get rid of these and enlarge them when necessary. Before we do, though, let’s begin by identifying the criteria we need. We know that the toy factory produces two-engine planes and four-engine planes. We can see on each of our graphs that the 𝑥-axis represents the number of two-engine planes, whilst the 𝑦 represents the number of four-engine planes. So let’s define 𝑥 to be the number of two-engine planes and 𝑦 to be the number of four-engine planes.

Let’s now create an expression that describes the number of hours required in the assembly department by the two planes. We’re told that the two-engine plane requires six hours, whilst the four-engine plane requires eight. So the total number of hours all together required in the assembly department must be six 𝑥 plus eight 𝑦. Now, in fact, we’re told that the maximum number of working hours per week is 120 in that department. So we can use inequalities to describe this. We can say that per week six 𝑥 plus eight 𝑦 must be less than or equal to 120.

In a similar way, we’re told that each two-engine plane requires one hour in quality control, whilst each four-engine plane requires two hours. So the total number of hours required in quality control is 𝑥 plus two 𝑦. But we’re told the maximum number of working hours per week in this department is 25. So 𝑥 plus two 𝑦 must be less than or equal to 25. Our job will now be to plot these on a coordinate plane. Of course, both 𝑥 and 𝑦 represent natural numbers. They’re the number of two-engine and four-engine planes. This must mean they must be both greater than or equal to zero. This, in turn, means that we’re only interested in the first quadrant of the coordinate plane.

The region represented by the inequality 𝑥 is greater than or equal to zero is the set of all points that lie above and on the 𝑥-axis. Similarly, for 𝑦 is greater than or equal to zero, we want the set of all points that lie to the right or on the 𝑦-axis. Let’s now plot the line six 𝑥 plus eight 𝑦 equals 120. We’re going to add a solid line to our diagram since our inequality is weak. We might write this in slope–intercept form. Subtracting six 𝑥 and dividing through by eight, the equation of the line we’re going to plot is 𝑦 equals negative three-quarters 𝑥 plus 15. And this line looks like this. We see it passes through the 𝑦-axis at 15 and the 𝑥-axis at 20.

We can, of course, calculate that value by setting 𝑦 equal to zero and solving for 𝑥. We’re interested in the set of points that satisfy the inequality six 𝑥 plus eight 𝑦 is less than or equal to 120. Now, in fact, that’s going to be the set of points that lie below the line. We can convince ourselves this is true by choosing the point with coordinates zero, zero. Six times zero plus eight times zero is indeed less than 120.

So with that in mind, let’s plot the second line. Subtracting 𝑥 and dividing by two, and we need to draw the line 𝑦 equals negative a half 𝑥 plus 12.5. This once again is a solid line because our inequality is weak. It will pass through the 𝑦-axis at 12.5 and the 𝑥-axis at 25. Once again, the set of points we’re interested in lie below this line. We can convince ourselves this is true by looking at the point zero, zero. Zero plus two times zero is less than 25.

So we have our lines and we know that each of these points must lie above and to the left of the 𝑥- and 𝑦-axes, respectively. And so we can shade the region shown on the diagram. With that in mind, we can now identify the correct graph that represents the number of planes produced per week. In our earlier list of multiple choice options, that was option (B). The shaded region represents the number of planes produced per week.

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