Minimum and Maximum Pollution
Date: 04/26/2000 at 00:36:24 From: Jenny Dailey Subject: Pre-calculus I have been trying to do this homework problem for a week now and the furthest I have gotten is drawing the picture. Suppose that pollution at a particular location is based on the distance from the source of the pollution according to the principle that for distances greater than or equal to 1 mile, the concentration of a particular matter (in parts per million, ppm) decreases as the reciprocal of the distance from the source. This means that if you live 3 miles from a plant emitting 60 ppm, the pollution at your home is (60/3) = 20 ppm. On the other hand, if you live 10 miles from the plant, the pollution at your home is (60/10) = 6 ppm. Suppose that two plants 10 miles apart are releasing 60 and 240 ppm, respectively. At what point between the plants is the pollution a minimum? Where is it a maximum? I have thought the obvious answer is next to the 60 ppm plant, but that makes no sense. I am supposed to find a formula to enter into my graphing calculator to help me find the maximum and minimum, but I am having no luck at all. Help me, please! :)
Date: 04/26/2000 at 17:35:17 From: Doctor Paul Subject: Re: Pre-calculus Let's begin by looking at a picture of what's going on here: We'll assume that the 60-ppm plant is on the left but it really doesn't matter. Let's say that you live x miles from the 60-ppm plant. Then you must live (10-x) miles from the 240-ppm plant. You've essentially divided up the 10 miles between plants into 2 sections. The first section is of length x. The second section must be of length 10-x (because the two sections have to add up to 10.) At your residence, you are receiving pollution from two plants. The pollution you receive from the 60-ppm plant is 60/x. The pollution you receive from the 240-ppm plant is 240/(10-x). The total pollution at your residence will be the sum of these two individual pollutions, so the total pollution at your residence will be: 60/x + 240/(10-x) But this formula is only good for values of x between 1 and 9. Here's why. Let's think about the problem. The problem never says anything about the amount of pollution if you're less than or equal to 1 mile from the plant. I think it's safe to assume that everyone within a mile of the plant gets the same amount of pollution. How do we know what this value is? The formula tells us the amount of pollution when we are exactly one mile from the plant (as pollution is constant within a 1-mile radius of the plant, this will also be the pollution value for everyone living less than or equal to 1 mile from the plant.) This is just the number of ppm that the plant emits divided by one, which is simply the number of ppm emitted by the plant. It should be obvious that to maximize the pollution, you want to be as close to the 240-ppm plant as possible. But as you get closer to the 240-ppm, the amount of pollution from the 60-ppm plant is going down (because you're moving away from it.) Notice that if you pick x = 9, you're getting a maximum amount of ppm from the 240-ppm plant and that you're getting 60/9 ppm from the 60-ppm plant. So at x = 9, your total pollution is 240/1 + 60/9. If you move to x = 9.5, you're still getting 240 ppm from the 240-ppm plant (because you're only a half mile away from it), but your value from the 60-ppm plant has been reduced from 60/9 to 60/9.5. We can use a graphing calculator to graph the function defined above and verify that x = 9 is, in fact, the maximum value when x ranges from 1 to 9. I can't show you a graphing calculator output, but I can show you Maple output. Maple is math software that performs operations similar to graphing calculators. Here's what Maple says the graph looks like: You can clearly see that the maximum occurs at x = 9. Now for the minimum. You can see from the Maple output that the minimum occurs somewhere between 2 and 4. You can use the trace feature on your graphing calculator to find the approximate point where the graph bottoms out. Or, if you know Calculus, you can use the first derivative test to find the exact point where the minimum occurs. Take the derivative of the function, set it equal to zero and solve for x. I got Maple to do this. Here's what Maple says is the minimum point: You can verify that this is the minimum using the trace feature on your graphing calculator. A summary of our results: The pollution is a minimum when we are 10/3 miles from the 60-ppm plant and 20/3 miles from the 240-ppm plant. The maximum occurs when we are 9 miles from the 60-ppm plant and 1 mile from the 240-ppm plant. You can compute the actual pollution values at these spots by plugging the appropriate value for x into the equation given above. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/
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