Modeling Ocean Temperatures


Modeling Ocean Temperatures

The Problem

The problem: The monthly average ocean temperature in degrees Fahrenheit at Bermuda can be modeled by

f of x equals point 0 1 2 5 x to the fourth power minus point 6 4 8  x cubed plus 6 point 0 3 x squared minus 17.1 x plus  seventy-six point 4

where x =1 corresponds to January and x = 12 corresponds to
December. The domain of f is
x is between 1 and 12
(Source: J. Williams, The Weather Almanac 1995.)

This model will be examined by:

a). Graphing f in [1,12,1] by [50,90,10].
b). Estimating the absolute extrema and interpreting the results.

a). First, an appropriate Viewing Rectangle will be set. Press WINDOW and enter 1 for Xmin, 12 for Xmax, 1 for Xscl, 50 for Ymin, 90 for Ymax, and 10 for Yscl. Next press and enter the following for Y sub 1 :

Y sub 1 = f of x equals point 0 1 2 5 x to the fourth power minus point 6 4 8  x cubed plus 6 point 0 3 x squared minus 17.1 x plus  seventy-six point 4

Now press GRAPH and the graph shown below will appear.

Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
 


b). The absolute minimum point (the lowest point) of the graph represents the month when the oceans were the coldest. The absolute maximum point (the highest point) represents the month when the temperatures were the warmest.

To locate the approximate absolute minimum point, look for the lowest point on the graph. To find the coordinates of this point, press 2nd TRACE[CALC] and select 3. Use the LEFT ARROW KEY to move the cursor until it is slightly to the left of the point and press ENTER. Now use the RIGHT ARROW KEY to move the cursor so that it is slightly to the right of the point and press ENTER. Press ENTER, and then ENTER again.

Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator

Notice that the absolute minimum point is approximately (2.01, 61.5).

Thus, in February (x = 2) the coldest average ocean temperature is approximately 61.5 degrees Fahrenheit.

To locate the approximate absolute maximum point, look for the highest point on the graph. To find the coordinates of this point, press 2nd TRACE[CALC] and select 4. Use the RIGHT ARROW KEY to move the cursor until it is slightly to the left of the point and press ENTER. Now use the RIGHT ARROW KEY to move the cursor so that it is slightly to the right of the point and press ENTER. Press ENTER, and then ENTER again.

Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator
Screen Picture of TI-83 Calculator

Notice that the absolute maximum point is approximately (7.61, 82.01). Thus, in August (xapproximately equal to8) the warmest average ocean temperature is approximately 82 degrees Fahrenheit.


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