Calculus Student
Franklin & Marshall College
Lancaster, PA 17604
Dear Calculus Student:
In these times of great concern over the health of our planet, there are few
people who do not know about the overwhelming problem of waste disposal. We
entrepreneurs at Eco-Sludge are not immune to this concern; in fact, we are
directly responsible for disposing of the waste by-products for the larger part
of the state of PU. Unfortunately, our former board of managers, which has now
been forcibly retired by the state's Environmental Agency, spent the greater
part of its energies on improving cost-effectiveness to the grave detriment of
safety and health. As a result, we are now faced with a expensive and what is worsedangerous cleaning project.
One of the first tasks facing us is estimating the total damage done. Your
keen and piercing professor, Dr. "Death" Crannell, told me of your resounding
success with the Jumping Gelatin Factory, and so I am being so bold as to ask
you for help.
Our sludge is stored in large containers buried beneath the ground. These
containers were manufactured with ease-of-construction in mind, and are really
nothing more than parabolic holes in the ground, lined with concrete and covered
with lids. (The problem is that the concrete cracks because of the changes in
temperature due to our erratic climate. The sludge has been seeping out into
the ground and making its way to the water supply. Needless to say, this is a
horrendous state of affairs for our local fishers as well as for any inhabitant
wishing to drink water from the faucet).
There are several measurements concerning the storage containers that we can
make with ease. Obviously, we can determine the diameter and the radius of the
vat's opening with a large tape measure. We have a plumbing rod that we can use
to determine the depth of the container and also the depth of the sludge (this
is done much the same way you check the oil in your car: by pulling the stick
out and seeing how much of it is wet). According to specifications, each vat
should be 40 feet across at the top and 40 feet deep. In reality, we have found
that the measurements of the vats vary significantly, although the general
parabolic shape does not. And, to no one's surprise, there is neither rhyme nor
reason to the depth of sludge found in a particular vat.
What we can not easily determine is what we most need to know: the quantity
(volume) of the sludge in any given vat. We are fairly certain that there is,
however, a mathematical way of deriving this formula. We would be most grateful
if you could help us with this, either in the standard (40'x40') case, oreven
betterin the more general case.
Our report to the trustees must be mailed by November 6; could you please get
the answer to us before that date? We await your reply with eagerness.
|
