Video Transcript
A company manufactures a product in two different production plants, 𝑃 sub one and 𝑃 sub two. 𝑃 sub one produces 20 units per month, and 𝑃 sub two produces 60 units per month. Each month, the company supplies at least 40 units of this product to customer one and at least eight units to customer two. Customer one is supplied with 20 percent and 80 percent of their needs from 𝑃 sub one and 𝑃 sub two, respectively, while customer two is supplied with 40 percent and 60 percent of their needs from 𝑃 sub one and 𝑃 sub two, respectively. Which part of the following graph represents the number of units supplied to both customers each month?
And then we have a graph with several regions, 𝐴 through 𝐾, labeled. Before we can identify which region or regions satisfy this information, let’s begin by representing it algebraically. We’re told that the company manufactures some product. Now it’s the same product, but it comes out of different production plants, 𝑃 sub of one and 𝑃 sub two. We’re then told information about how the company supplies these products to two different customers. Specifically, we’re told that they supply at least 40 units to customer one and at least eight to customer two. Let’s clear some space and define our variables.
We might notice that on the graph the number of units supplied to customer one is the label for our 𝑥-axis. So let’s define 𝑥 to be the number of units supplied to customer one. Similarly, the 𝑦-axis is labeled as the number of units supplied to customer two. So we’re going to define 𝑦 to be the number of units supplied to this customer. We’re told that each month the company supplies at least 40 units to customer one. This means they could supply 40 or more units. So we let 𝑥 be greater than or equal to 40. Similarly, we’re told that they supply at least eight units to customer two. So we’re going to let 𝑦 be greater than or equal to eight.
Then we move on to the second paragraph. Specifically, we’re interested in the part that tells us what percentage of each product comes from each production plant. We’re told that 20 percent of customer one’s needs come from 𝑃 sub one, whilst 40 percent of customer two’s needs comes from 𝑃 sub one. 20 percent as a decimal is 0.2. So we could work out the total number of units that customer one gets from 𝑃 sub one by finding 0.2 of 𝑥. That’s 0.2𝑥. In a similar way, 40 percent as a decimal is 0.4. So we could find the total number of units that customer two receives from the first production plant by calculating 0.4𝑦.
And then that’s really useful because we actually know the total number of units produced per month by 𝑃 sub one. It’s 20. This means that the sum of the number of units that customer one and customer two receive from the first production plant must be less than or equal to 20. If we multiply through by five, we see this is equivalent to saying 𝑥 plus two 𝑦 is less than or equal to 100. Let’s repeat this using information about the percentage of the needs each customer receives from 𝑃 sub two.
We’re told customer one receives 80 percent. We can find the total number by multiplying 0.8 then by 𝑥, the number of items that customer one receives. Then we’re told 60 percent of customer two’s needs are taken from 𝑃 sub two. That’s 0.6𝑦. Then we’re told that 𝑃 sub two produces 60 units per month. So the total number of units that customer one and customer two receive from this second production plant, that’s 0.8𝑥 plus 0.6𝑦, can have a maximum value of 60. So we can say 0.8𝑥 plus 0.6𝑦 is less than or equal to 60. Once again, we can multiply through by five, and we get four 𝑥 plus three 𝑦 is less than or equal to 300.
We now have four inequalities, and we need to link these to the graph we’ve been given. Let’s begin with the inequality 𝑥 is greater than or equal to 40. To plot this region on a graph, we begin by drawing the line 𝑥 equals 40. That’s a vertical line that passes through 40 on the 𝑥-axis. On our diagram, we can see that that’s the green line shown. Since the inequality is weak, in other words, 𝑥 is greater than or equal to 40 rather than just greater than 40, we draw a solid line to represent this inequality.
So which side of the line are we interested in? Which side of the line represents the region 𝑥 is greater than or equal to 40? All the values of 𝑥 that are greater than or equal to 40 are to the right of this line. So what we’re going to do to make it easier to follow is eliminate the region we are not interested in. We shade the left-hand side of this line to eliminate that region. And this is great because this eliminates several parts of our graph. Specifically, 𝐴, 𝐵, 𝐸, and 𝐻 all lie in the region that we’ve stated do not satisfy the inequality 𝑥 is greater than or equal to 40. So these cannot be the regions we’re interested in at the end.
Let’s repeat this process for the inequality 𝑦 is greater than or equal to eight. We’d begin by drawing the line 𝑦 equals eight on the diagram. This time, that’s a horizontal line that passes through the 𝑦-axis at eight. So it’s the purple line shown. We need to identify which side of the line satisfies the criteria 𝑦 is greater than or equal to eight. Well, all values of 𝑦 that are larger than eight lie above this line. So once again, let’s shade the side of this line that we’re not interested in. We shade the region that lies below the line 𝑦 equals eight as shown.
Now at this point, it might be worth observing that there were two further inequalities we could have written. 𝑥 and 𝑦 represent numbers of units. This must mean that they are natural numbers. In other words, they cannot be negative. However, we already identified that 𝑥 must be greater than or equal to 40 and 𝑦 must be greater than or equal to eight. By definition then, 𝑥 and 𝑦 are already recognized to be positive through these two inequalities.
Let’s now move on to our third inequality, 𝑥 plus two 𝑦 is less than or equal to 100. Subtracting 𝑥 from both sides and then dividing through by two, and we see that we can plot the line that will help us represent this inequality by plotting the line 𝑦 equals 50 minus one-half 𝑥. This has a 𝑦-intercept of 50 and a slope of negative one-half. So we can see that this is the red line on our diagram. It’s a solid line because it represents a weak inequality. And we now need to decide which side of the line represents 𝑦 is less than or equal to 50 minus one-half 𝑥.
We could check by choosing a point on either side of the line. The point zero, zero is often a really useful point to choose. We substitute zero into the inequality and check whether it satisfies it. Zero is less than or equal to 50 minus a half time zero. In other words, zero is less than or equal to 50. Well, yes, zero is less than 50. So the side of the line that the point zero, zero lies on is the side of the line we’re interested in. As before, we’ll shade the other side of the line to disregard that region. On our diagram, we shade all points above the red line.
And so we now see that we’re interested in two possible regions, either 𝐹 or 𝐺. Let’s consider our fourth and final inequality and establish which one it is. We begin by rearranging and making 𝑦 the subject. And we see we need to plot the line 𝑦 equals 100 minus four-thirds 𝑥. This has a 𝑦-intercept of 100 and a slope of negative four-thirds. And so it’s the blue line.
Then we need to identify which side of the line satisfies our inequality. Once again, we’ll consider the point zero, zero. If we substitute zero, zero in, we get zero is less than or equal to 100. Well, yes, zero is less than 100. And so the side of the line that satisfies our inequality is the side containing the point zero, zero. We’ll shade the other side to disregard this from our region. And when we do, we see we’re left with a simple quadrilateral representing the region that satisfies our criteria. Region 𝐹 satisfies all four inequalities that we wrote. And so the part of the graph that represents the number of units supplied to both customers each month is 𝐹.
