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The attached file is a posting from another listserve that I thought
would be of interest to Calculus teachers on this list.
Perhaps even more interesting than the particular problem being
discussed is the impact of the "packaging" of a problem on student
attitude towards the validity of the problem (in a wider sense, the
validity of the study of mathematics itself). This should be of
interest to teachers of pre-service math teachers.
I plan to use this material in my classes (high school calculus) both
ways--as the problem and as a discussion of problem packaging.
I am also forwarding another related post.
Syrilda
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Date: Fri, 06 Feb 1998 11:25:09 -0500
From: "penn.howard" <penn.howard@mathnt1.sma.usna.navy.mil>
Subject: RE: The Lifeguard
To: "'Losse@sc.maricopa.edu'" <Losse@sc.maricopa.edu>,
"'Calc Reform'" <calc-reform@MATH.AMS.ORG>,
"'Jerry Uhl'" <juhl@ncsa.uiuc.edu>
Cc: "'Graph-ti'" <GRAPH-TI@lists.ppp.ti.com>
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Here is my version of the problem. There are a couple of interesting
things that come out of the project. The critical point turns out to be
independent of the distance down the beach and therefore if the target
point is less than that distance down the beach the fastest route is the
straight line. If you then express the distance in terms of the angle,
you find a formula for the optimal angle as a function of the two speed
so that you only need to solve the problem once and you have it for all
values of the two distances.
Howard Penn
You are the commanding officer in an AVT. Your current position is 5
Nautical miles off the coast of enemy held territory. Your orders are
to attack a railway station which is 10 Nautical miles East along the
coast from the closest point of land. Your AVT can travel 7 knots on
water and 14 knots on land. You are to attack this station in the
shortest amount of time. See the diagram below.
DETAILS OF THE PROJECT
1. Set up a formula for the time, T, to reach the station as a function
of the distance from the closest point to the landing point(x). Plot
this function, T, for x in the interval (0,10). Estimate the value of x
which minimizes the total time. Solve using the derivative to
determine the find the value of x that will minimize the time needed to
reach the station.
2. Repeat the process for the distance to the station being 5 Nautical
miles and 2 Nautical miles. What do you notice about the best place to
land in these problems?
3. Now solve the problem in general. Let the speed on water be Vw, the
speed on land be Vl, the distance from the shore be a and the distance
up the shore be b.
4. Express as a function of x. Use this to find an angle which will
minimize the time needed to reach the station. What constants does this
angle depend on?
>----------
>From: Jerry Uhl[SMTP:juhl@ncsa.uiuc.edu]
>Sent: Friday, February 06, 1998 4:01 AM
>To: Losse@sc.maricopa.edu; Calc Reform
>Cc: Graph-ti
>Subject: Re: The Lifeguard
>
>At 2:10 PM -0700 2/5/98, John Losse wrote this and more:
>
>>Here is an extension of a classical-type problem which may be
>>interesting for calculus students.
>>
>>An example of the classical problem was used in an article on the TI-92
>>by Charles Vonder Embse and Vernon W Yoder in the January, 1998
>>Mathematics Teacher:
>>
>>A lifeguard is 150 meters away from the point on a straight shoreline
>>closest to a swimmer who is 60 meters offshore. If the lifeguard can
>>run at 8 m/sec and can swim at 2 m/sec, where should the lifeguard jump
>>into the water to minimize the total time to reach the swimmer?
>>
>
>>
>>c) Can you coax the answer to these general questions out of your TI-92,
>>your MAPLE, your Mathematica, etc? Is it worth it?
>>
>Comment by Jerry Uhl:
>
>If the lifeguard actually solved the problem before jumping in to save the
>swimmer, then the swimmer would surely perish.
>
>This puts this problem in the category of problems interesting to math
>teachers like us, but not to many students. They see it as a classroom
>problem. In other words,this problem presents to the students a disconnect
>between math and reality. And that's not good for us or for our students.
>
>Fortunately this problem can be at least partially salvaged by switching to
>a more realistic context:
>Here's my quick attempt.
>
>An oil rig is 0.5 miles (perpendicular distance) away from a point on a
>straight shoreline.
>The supply center for the oil rig is 2.7 miles down shore from this point.
>The amphibian supply truck makes a daily trip from the supply center to the
>oil rig. The amphibian makes 10 miles per gallon on the road down the shore
>line and makes 2.1 miles per gallon in the water.
>What route minimizes fuel?
>
>Maybe some else can suggest a better fix.
>
>
>
>
>
>
>
>----------------------------------------------------------------------
>Jerry Uhl juhl@ncsa.uiuc.edu
>Professor of Mathematics 1409 West Green Street
>University of Illinois Urbana,Illinois 61801
>Calculus&Mathematica Development Team
>
>http://www-cm.math.uiuc.edu
>http://www-cm.math.uiuc.edu/dep
>
>
>
>
>
>
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