#### Math 142 - Calculus II - Fall 2011 – Prof.
Janeba

### Project #1

GOALS:

- To gain experience dealing
with unusual problems for which there may not be a single right method of
solution. The answer may take many forms and needs to be explained
in detail through the use of words, numbers, formulas, and graphs.
- To gain experience explaining
mathematical results in English.
- To practice working with
others on a mathematical problem.
- To become familiar with the
members of the group with whom you will be working on these projects.

Harold Hadley is very lazy and very rich. Oh, and he loves to ride his bicycle. Yes, these may seem contradictory, but there
it is.

As Harold rides his bicycle through Salem, he often encounters this problem: As he rounds a certain bend, he sees a car
ahead stopped at a red light. The car is
blocking Harold’s path, so he needs to slow down, but being lazy, he hates to
slow down any more than necessary – for then he will have to expend energy to
regain his lost speed.

Now the traffic signal timing and Harold’s cycling habits
(including speed) are so consistent, that the situation is often *precisely* this: His velocity is 20 m.p.h. and he sees the car
when his front wheel is 100 feet behind the car’s rear bumper. Also, these cars seem to accelerate pretty
consistently at 7 ft/sec/sec when the light turns green, and the light changes just
3 seconds after Harold sees the car.

You may wonder why Harold doesn’t just get in the habit of
coming around this bend at a slower speed, but often the light is green, there
is no stopped car, and he need not slow at all.

So Harold is commissioning your consulting team to answer
this question: If he brakes at a
constant deceleration starting at the instant that he sees the stopped car,
what should that deceleration be so that he doesn’t hit the car but retains as
much of his speed as possible? Oh, and
what would happen (give great mathematical detail) if he didn’t brake at all?
You should neglect any non-braking deceleration that would occur due to air
resistance or other friction. Granted, these
may seem silly questions to invest so much time on, but Harold is rich, and can
hire people to do what he wants. By the
way, it’s probably best not to bring up to Harold the vanity of assuming that
he can brake at precisely a given deceleration.
Humor him.

Your group, wanting to earn more of Harold’s money in the
future, will want to make him happy. This
means that of course you’ll want the answer to be right, and you’ll want to
make sure of that by carefully checking, preferably in multiple ways. [Your grade cannot be higher than a C if the
answer is not exactly correct – Prof. J.]
Furthermore, you want to persuade Harold that you weren’t just lucky in
finding the right answer, so you’ll need to give a careful report of how you
got it. Harold *did* take calculus in college, though he’s forgotten many of the
concepts, and will need some reminding.
He’s still good at algebraic calculation, though. So you’ll want to write up a detailed report explaining
how you set up the problem, why your setup is correct, and how you solved
it. Algebraic simplification doesn’t
need lots of explanation, but you will need to give enough detail that Harold
can reproduce your calculations without any guessing on his part. You’ll also want to look at the overall
question and situation, describe just exactly what your solution will
accomplish, and see if you can offer some tactful advice to Harold that he
might appreciate. [This is how you earn
a grade higher than a C.]

Consulting firms will be required to make a private group
presentation of preliminary results to Mr. Hadley’s
consultant evaluator, Prof. Janeba, on either Wed. Sept. 7, or Thurs.
Sept. 8. Please make an appointment with Prof. Janeba - there are limited
appointments available on each day. Come to your appointment with at least
a fairly detailed plan for finding the solution, if not the solution itself, already
worked out **and in written form**.

Final reports are to be printed on 8.5x11 inch white paper sheets
bound with
a single staple, submitted to the consultant coordinating office
(Ford 216) no
later than Tuesday,
Sept. 13, at 4:00 p.m...
Be sure that your report is self-contained (in case Mr. Hadley forgets
what the question was), and its qualities of completeness, clarity, and
correctness reflect your best abilities. It is
perfectly acceptable to write in complex formulas by hand, although
groups may find it worthwhile to expend a small effort getting the word
processor to type simple formulas and symbols. NOTE: anything
simply
shoved under the door after 4 pm that day will be
assumed to have been too late!

A Word to the wise: This project is worth about 10% of your course
grade.
Please make sure that you are doing the best that you can. Please
carefully read
all the instructions and resources provided.

*Last Modified August 30, 2011.*

Prof.
Janeba's Home Page | *Send comments or questions to:* mjanebawillamette.edu

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