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Whelk-Come to Mathematics: Using Rational Functions
to Investigate the behavior of Northwestern Crows
Part Four - Work to a Conclusion
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The amount of work in dropping a whelk to break it open
depends on the height of the drop and the number of times
a whelk has to be dropped.
Work = Height * Number of Drops
W = H * N
If you've been working with a data set, the data should
automatically be filled in below. Otherwise, enter your
own data or select one of the two other data sets.
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The relationship between the height of the drop and
the number of drops can be used to investigate the work.
Using the method of transforming the data and using
linear regression, the relationship between H
and N for the sample peanut data is:
The equation for the amount of work for the sample
peanut data is:
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Activity 1
Use the function
grapher or your own graphics calculator to find
the height corresponding to the minimum work.
- What is the height at which the minimum
work occurs? How do values for the work near
this height compare to the minimum work?
- How does the location for the minimum work
you found using the equation, compare to the
value you observed from the data? Which finding
do you think should be reported and why?
- What is true about the work for large heights?
Give an explanation for your observations.
- What are the asymptotes for the work equation?
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Both the expression for the number of drops in terms
of the height and the work in terms of the height are
rational expressions. Rational expressions are those
that can be expressed as the quotient of two polynomials.
Rational expressions can be written in several different
forms. Three common ways of writing a rational expression
are show below for the same rational expression:
Standard Form: 
Factored Form: 
Proper Fraction Form: 
Each form of a rational expression provides different
information and insight into the nature of the function.
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Activity 2
Rewrite your expressions
for N and W in each of these forms.
Graph each function and determine features of
the graph such as asymptotes, zeros, or the general
shape.
- What information can you determine about the
function from each form?
- Examine the proper fraction form. What does
this form tell you about the amount of work
for very large heights? for very small heights?
Hint: What is the major contributing factor
to the amount of work required in each case?
- Write a definition for when a rational function
f(x) is written in proper fraction form
by completing the following lead–in sentence.
Do so by placing conditions on the polynomials
p(x), r(x) and q(x).
- If f(x) is a polynomial written in proper
fraction form using the polynomials p(x),
r(x) and q(x) such that
, then ...
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Reflection Questions
Examine your original work equation
and the different ways of expressing the same
expression for N and W.
- Some biologists hypothesize that the dropping
of whelk from a height of 5 meters by the
crows is an example of optimal foraging. Does
the Large Whelk data provide evidnece to support
or refute this claim? Give specific evidence.
- Work also depends on the weight of the object.
Northwestern crows drop only large whelk which
are fairly comparable in weight of about 8.8
grams.
- How would you alter the equation to
include the factor of weight?
- How is the height at which work is a
minimum affected if the factor of weight
is included?
- How was knowledge about rational functions
useful in finding a mathematical model for
the work involved?
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Credits and References
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