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
Based on the computed values of work for the individual data points, between what heights is the amount of work the smallest?
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:
Use 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?
- Compare the location for the minimum work you found using the equation to
the value for the minimum work 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
- What are the asymptotes for the work equation?
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:
Proper Fraction Form:
Each form of a rational expression provides different information and insight into the nature of the function.
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
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 evidence to support or refute this claim? Give specific
- 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?