Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

```

```
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.

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

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search