And The Winner Is?
Calculus 2, Spring 2002, Central
College
Project 1, Tom Linton
General projects advice sheet.
Guidelines for this project.
A horse race at Suluclac Park featured a close finish, with the top
three horses crossing the finish line at virtually the same time.
The race covered a distance of 1.5 furlongs. The only data remaining from
this race involves the velocities of the horses, in units of furlongs per
minute, and a movie, which is not really conclusive (and may have been
tampered with). The velocity data for the top three horses is represented
in various forms
-
one is given by a table of values;
-
one is given graphically;
-
one is represented algebraically by a formula.
The racing commission needs your help to decide (with a convincing argument
and a professional looking write-up) which of the three horses actually
crossed the finish line first. Unfortunately, not all members of
the commission understand calculus real well. You may assume that
they all
-
know the material in calculus 1,
-
know how to compute areas of common geometric figures (rectangles, triangles
and the like),
-
understand the idea that distance is equal to the area under the velocity
curve,
but the Fundamental Theorem of Calculus is beyond a few key members of
the commission. Any use of this theorem should be backed up with
both over and under estimates using rectangles. Several members of
the racing commission are easily impressed with graphical evidence, so
it may help to include pictures drawn with left rectangles or right rectangles
(or better yet a combination of both) which clearly show why certain rectangles
give over estimates, and why others give under estimates. Unfortunately,
the velocities of these horses are not monotone, so they increase and then
decrease etc. (you'll have to break the race up into pieces, were the velocities
increase or decrease over each piece). Finally, the chairman of the racing
commission is a mathematician whom will certainly discredit any report
that doesn't use the Fundamental Theorem of Calculus at least once. As
a side note, there is really no need to use fnInt on your
calculators, since you now know the Fundamental Theorem, and the only formula
given is a polynomial. You should look up the definition of a furlong,
and include that in your report.
The first horse, MidRangeMover, is known for strong perfomances
near the middle of most races. The velocities for MidRangeMover are
given in the tables below. Again, the times are measured in minutes
and the velocities in furlongs per minute. This data can be found
in the spreadsheet file (MS EXCEL) G:\Lintont\math132\horseproject.xls.
Since MidRangeMover's velocities are only known at certain times, it is
impossible to determine the exact time this horse crossed the finish line,
but you can certainly give under estimates and over estimates of this horse's
finishing time. You should NOT attempt to find a formula for this horse's
velocity function.
| Time |
0.00 |
0.02 |
0.04 |
0.06 |
0.08 |
0.10 |
0.12 |
0.14 |
0.16 |
0.18 |
0.20 |
0.22 |
| Velocity |
0.000 |
0.009 |
0.022 |
0.039 |
0.060 |
0.084 |
0.112 |
0.143 |
0.178 |
0.215 |
0.255 |
0.298 |
| Time |
0.24 |
0.26 |
0.28 |
0.30 |
0.32 |
0.34 |
0.36 |
0.38 |
0.40 |
0.42 |
0.44 |
0.46 |
| Velocity |
0.343 |
0.391 |
0.440 |
0.491 |
0.542 |
0.595 |
0.648 |
0.702 |
0.755 |
0.808 |
0.860 |
0.912 |
| Time |
0.48 |
0.50 |
0.52 |
0.54 |
0.56 |
0.58 |
0.60 |
0.62 |
0.64 |
0.66 |
0.68 |
0.70 |
| Velocity |
0.962 |
1.010 |
1.056 |
1.100 |
1.142 |
1.181 |
1.217 |
1.250 |
1.279 |
1.306 |
1.328 |
1.347 |
| Time |
0.72 |
0.74 |
0.76 |
0.78 |
0.80 |
0.82 |
0.84 |
0.86 |
0.88 |
0.90 |
0.92 |
0.94 |
| Velocity |
1.362 |
1.373 |
1.380 |
1.384 |
1.384 |
1.379 |
1.372 |
1.360 |
1.345 |
1.327 |
1.306 |
1.282 |
| Time |
0.96 |
0.98 |
1.00 |
1.02 |
1.04 |
1.06 |
1.08 |
1.10 |
1.12 |
1.14 |
1.16 |
1.18 |
| Velocity |
1.255 |
1.225 |
1.194 |
1.160 |
1.125 |
1.088 |
1.051 |
1.012 |
0.974 |
0.935 |
0.896 |
0.858 |
| Time |
1.20 |
1.22 |
1.24 |
1.26 |
1.28 |
1.30 |
1.32 |
1.34 |
1.36 |
1.38 |
1.40 |
1.42 |
| Velocity |
0.821 |
0.785 |
0.751 |
0.718 |
0.688 |
0.659 |
0.634 |
0.611 |
0.591 |
0.575 |
0.562 |
0.552 |
| Time |
1.44 |
1.46 |
1.48 |
1.50 |
1.52 |
1.54 |
1.56 |
1.58 |
1.60 |
1.62 |
1.64 |
1.66 |
| Velocity |
0.546 |
0.544 |
0.545 |
0.551 |
0.560 |
0.573 |
0.590 |
0.611 |
0.635 |
0.663 |
0.694 |
0.729 |
| Time |
1.68 |
1.70 |
1.72 |
1.74 |
1.76 |
1.78 |
1.80 |
1.82 |
1.84 |
1.86 |
1.88 |
1.90 |
| Velocity |
0.766 |
0.806 |
0.849 |
0.895 |
0.942 |
0.991 |
1.042 |
1.093 |
1.146 |
1.199 |
1.253 |
1.306 |
The second horse, UpandDown, known for quick bursts of speed,
had the velocities represented graphically below. Here is a hint
on how to calculate the exact time that UpandDown finishes the race
(using only geometry, so the entire racing commission will understand).
Calculate the area under the velocity graph from t = 0 until t = 1.3 minutes
(hopefully this is less than 1.5 furlongs). Now assume that UpandDown
runs for t minutes more. The additional area will be composed
of a rectangle and a triangle (the dimensions of both will involve the
unknown variable t). Adding this variable amount of extra area to
the area under the curve from 0 to 1.3, will give a formula for the exact
distance that UpandDown travels in the first 1.3 + t minutes. Set
this equal to 1.5 furlongs and solve for t. The Excel
file referred to above, contains a graph similar
to this plot (the plot in the Excel file can be clicked on and dragged
to make it larger or smaller and then printed) or may be found at G:\Lintont\math132\horseproject.xls.
A full size version of this plot is also included in this handout. You
can also click on the image below to see the gif image file alone.
The third horse, EndBurner, known for strong finishes, had a velocity
of 4.8t - 7.2t2 + 3t3 furlongs per minute, t minutes
into the race. This velocity function is NOT monotone for the entire
race (so cut it up into monotone pieces) and is relatively straightforward
to antidifferentiate (but recall the warning about using only the
Fundamental Theorem above). Over and under estimates for this horse
can be found which are close enough to the true values to decide which
horse won the race, but using just left rectangles or just right rectangles
for the entire length of the race won't cut it (there is no guarantee that
a right or left hand sum is an over or under estimate, since the velocity
function is NOT monotone). You will have to combine some left and
some right rectangles. Of course, for those members of the racing commission
that do understand calculus, you shoud also use the Fundamental Theorem
to determine the exact time that EndBurner crosses the finish line. Finally,
finding the spots where EndBurner's velocities switch from increasing to
decreasing, using calculus, will greatly impress the commisioner. A plot
of this velocity graph is included, in case you want to draw in some rectangles,
or something else for your report. There is also a version of this velocity
plot in the Excel file referred to earlier.
You may wish to use Mathematica to plot things as well.
Here is a movie of the race (on line only), which may have been
altered by one of the horse's owners (so it cannot be relied upon in your
report). It's hard to tell which horse wins from the movie, so you should
type up (using a word processor and some hand drawn or computer generated
graphs, tables, etc.) a clear explanation to the racing commission that
determines which horse won the race. Your report is due in the commissioner's
office (Central Hall 312 B) by February 14 at noon.
The Photo Finish
