Solved: Comparing Models A cup of water at an initial temperature of is placed in a room

Chapter 3, Problem 95

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Comparing Models A cup of water at an initial temperature of is placed in a room at a constant temperature of The temperature of the water is measured every 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form where is the time (in minutes) and is the temperature (in degrees Celsius). (a) The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points and (b) An exponential model for the data is given by Solve for and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use the graphing utility to plot the points and observe that the points appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. This resulting line has the form Solve for and verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of the -coordinates of the revised data points to generate the points Use the graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. The resulting line has the form Solve for and use the graphing utility to graph the rational function and the original data points. (e) Why did taking the logarithms of the temperatures lead to a linear scatter plot? Why did taking the reciprocals of the temperatures lead to a linear scatter plot?