Why Is Alaska So Cool?
Activity 6 Answer Key
Activity 6: Distance Intensity Investigator
Answer Key
1. What is your regression equation and correlation coefficient? Record these in your daily log
as well. Use labels or descriptions appropriately. Include the equation generated by the TI-83
as well as that generated by Excel.
Using sample data on spreadsheet attached, the regression equation is
y = 0.1793x-1.8417 and r2 = 0.9892. Rounded to the nearest integer, the regression
equation would be y = 0.1793x-2, so it appears to be an inverse square equation.
2. a. According to your model (regression curve) and using analytical methods described in
other activities, what would be the light intensity at 3 m? At 0.5 m?
Your answers will vary depending on the data you were able to collect. If you use
the sample data, light intensity at 3 m would be 0.0237 mW/cm2; at 0.5 m would be
0.6427 mW/cm2. The value you determined from your data for the 3 m distance
should show a much lower intensity than your lowest actual reading, and the 0.5 m
reading should be higher than your actual highest sample.
b. Which of the methods did you use to determine these theoretical values (e.g., evaluate a
function, creating a table of values, tracing graphs, estimating from the printed graph)?
The above was determined by evaluating the function. You may have used the trace
method, creating a table of values method. If you use the trace method, you will
note that it is necessary to increase the resolution (that is, to zoom in) of the graph in
order to get closer and closer to the exact values you seek.
3. Did you accurately predict the type of equation and graph that would best represent the
intensity of light as the distance form the light source increases? If not, what type of equation
did you predict?
Typically, the power regression is the best mathematical model for the data collected
in this investigation.
4. Explain why the polynomial regressions to the third and fourth powers (cubic and quartic)
were not the “best fit” regressions, even though they may have very accurately matched the
data that was collected.
Other models may also work well over the samples collected, and some may even
have a better r2 correlation. Nonetheless, as more and more data might be collected,
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Why Is Alaska So Cool?
Activity 6 Answer Key
at farther and nearer distances, it would become apparent that the overall best
predictor is the power regression.
So, how do you know which type of equation would work best from the beginning
of the process?
Past experience with mathematical modeling helps the most in figuring this
question. As close as the link is between mathematics and science, there is no known
connection that automatically invokes a particular equation for a set of data.
In other words, the process is trial-and-error. One guesses at an equation, and then
tests it against the real world: Does the mathematical model work well enough, or
not?
For example, the astronomer Johannes Kepler was first to recognize and describe the
true path of planets around the sun. He took a great part of his lifetime in pursuit of
the motion of the planets. The real world data he used was accurate and precise
enough, but no one had yet sorted out the actual shape of orbits. He finally came to
the stunningly simple conclusion that revolutionized cosmology: An ellipse (oval)
serves as the mathematical model for planetary orbit, with the sun situated at one
focus.
The process he used to find the model was strictly scientific trial and error: He began
with a model as a hypothesis and tested it for its ability to predict actual planetary
position. When his model hypothesis proved inadequate to predict well enough, he
adjusted with a new model as a hypothesis. He repeated this countless times until
the truth of the matter was discovered. Indeed, there are almost one thousand pages
in a museum still existing holding his actual calculations trying to find an adequate
model!
You may wish to explore the following web site to learn more about Kepler and the
solutions he derived.
http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Kepler.html
5. Since this relationship is common among many real world phenomena (including
gravitational attraction and magnetic forces) it is called the inverse square law. On one
occasion when the above investigation was done, the following regression was returned,
y=.179*x^-1.84, or y=.179*x-1.84. Round off the power to the nearest whole number. In what
way does this support the concept of the inverse square law?
When the power is rounded off to the nearer integer, the resulting equation is
y = 0.179x -2 = 0.179/x 2.
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Why Is Alaska So Cool?
Activity 6 Answer Key
This is precisely what the inverse square law states: one variable changes in inverse
proportion to the square of another variable.
But what does this mean? It means that as one variable increases the other decreases.
In this case, the intensity decreases as the distance increases.
Consider for a moment a teeter-totter or see-saw. Imagine two children, John and
Mary, of the same size sitting at equal distances from the see-saw's pivot. When one
child goes up, the other child goes down. Their distances from the ground are
opposite to each other, right? When Mary rises by one meter, John drops by one
meter. Their change in height, though opposite each other, are equal in magnitude
(size). If we consider their change in height from a perfectly balanced horizontal
position of the see-saw, the following equation could serve as a mathematical model.
y = John's height relative to the horizontal
x = Mary's height relative to the horizontal
Equation 1:
y = -x
Now suppose an adult (Dad) and child (Mary) are on the see-saw, but in order to
balance it out evenly, Dad sits closer to the pivot and Mary sits farther out from the
fulcrum. As they play, their distances from the ground are still opposite: when one
goes up, the other goes down. But, now if Dad drops one-half meter, Mary rises
more than one-half meter, maybe one or more meters, right? Their changes in height
are no longer equal in magnitude, but vary by some constant of change. If Dad
travels half as far as Mary, the model now may be described as follows.
z = Dad's height relative to the horizontal
x = Mary's height relative to the horizontal
Equation 2:
z = - 0.5x
where 0.5 is the constant of change, Dad's height change being one-half (0.5) of
Mary's.
This brief discussion shows that there are different ways to model events that have
reciprocal actions. The "inverse square law" is such a model, and has the added
benefit of describing well enough common paired relationships between important
phenomena in nature. Intensity vs. distance is one of those important pairs of
relationships.
Print a copy of the Excel graph to submit.
Graphs will vary, but should have appropriate titles, and labels on axes, as well as the
regression equation. See graph of sample data.
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