Sometimes we want to construct a model of some real-world phenomenon based upon data we've collected.
We're going to use software to do the heavy lifting for us.
Case Study #1: Do Florida boat registrations kill manatees?
The Data: The table below lists the number of Florida boat registrations and manatee deaths from 1977 to 2009.
Step #1: Make scatter diagram.
Below is a scatterplot of manatee deaths versus the number of boat registrations in Florida. Each point represents a particular year.
We will now enter the data into our TI-calculator and construct our own scatter diagram.
Step #2: Select the kind of function you want to use to model the data.
Step #3: Construct model. The equation for this line of "best-fit" for the manatee data is $$f(x)=0.129x-43.172.$$ We can use our TI-calculators to get the line of best fit...
Step #4: Interpret and analyze.
The equation for this line of "best-fit" for the manatee data is $$f(x)=0.129x-43.172.$$ Question: What does the slope of this line tell us?
Step #5: Use your model to make predictions.
The equation for this line of "best-fit" for the manatee data is $$f(x)=0.129x-43.172.$$ So, if we know in advance the number of boat registrations for the year, we can roughly predict how many manatee deaths will occur in that year.
Example: Suppose we expect 550 boat registrations this year. How many manatees can we expect to be killed?
Example: If we know 60 manatees were killed by boats, estimate how many boat registrations there were that year.
Case Study #2: Maximizing Profit.
An artisan chocolate company has been keeping records of the amount of chocolate (in tons) produced and sold along with their profit (in milllions of dollars) over several quarters. The data is below.
$$ \begin{array}{c|c} x & y \\ \hline 0.9 & -2.0\\ 1.9 & 0.4\\ 3.3 & 2.1\\ 0.1 & -7.7 \\ 6.7 & 2.9\\ 8.9 & -4.6\\ 10.0 & -8.2\\ 5.2 & 4.5\\ \end{array} $$
Step #1: Make scatter diagram.
Step #2: Choose the appropriate function type.
Step #3: Construct the model.
Step #4: Analyze and Interpret.
We now have a parabola of best fit which we may analyze:
$$f(x)= -0.487x^2 + 4.721x - 7.171$$ a. Vertex (if there is one).
b. Does the graph open upward or downward?
c. $x$-intercept(s) and $y$-intercept.
d. Domain and range.
e. Determine the $x$-values for which the function is positive and negative.
f. Determine the $x$-values for which the function is increasing and decreasing.
Step #5: Use the model to make predictions.
Our parabola of best fit:
$$P(x)= -0.487x^2 + 4.721x - 7.171$$ Question: Given the present business conditions, our company can make and sell 6 tons of chocolate this year. What profit can we expect this year?
$P(6)= -0.487(6)^2 + 4.721(6) - 7.171= 3.623$
Thus, we can expect a profit of about 3.6 million dollars this year.
Case Study #3: Predict human height from forearm length.