COMPARING MODELS A cup of water at an initialtemperature of is placed in a room at a

Chapter 5, Problem 97

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COMPARING MODELS A cup of water at an initialtemperature of is placed in a room at a constanttemperature of The temperature of the water ismeasured every 5 minutes during a half-hour period.The results are recorded as ordered pairs of the formwhere is the time (in minutes) and is thetemperature (in degrees Celsius).(a) The graph of the model for the data should beasymptotic with the graph of the temperature of theroom. Subtract the room temperature from each ofthe temperatures in the ordered pairs. Use a graphingutility to plot the data points and(b) An exponential model for the data isgiven by Solve for andgraph the model. Compare the result with the plotof the original data.(c) Take the natural logarithms of the revised temperatures.Use a graphing utility to plot the pointsand observe that the points appear tobe linear. Use the regression feature of the graphingutility to fit a line to these data. This resulting linehas the form Solve for andverify that the result is equivalent to the model inpart (b).(d) Fit a rational model to the data. Take the reciprocalsof the -coordinates of the revised data points togenerate the pointsUse a graphing utility to graph these points andobserve that they appear to be linear. Use theregression feature of a graphing utility to fit a lineto these data. The resulting line has the formSolve for and use a graphing utility to graph therational function and the original data points.(e) Why did taking the logarithms of the temperatureslead to a linear scatter plot? Why did taking the reciprocalsof the temperatures lead to a linear scatter plot?