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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/   
    
Associated Topics:
High School Calculus

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