A cup of water at an initial temperature of 78°C is placed in a room at a constant temperature of 21°C. 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 (t, T), where t is the time (in minutes) and T is the temperature (in degrees Celsius).
(0, 78.0°), (5, 66.0°), (10, 57.5°), (15, 51.2°), (20, 46.3°), (25, 42.4°), (30, 39.6°)
(a) Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points (t, T) and (t, T − 21).
(b) An exponential model for the data (t, T − 21) is T − 21 = 54.4(0.964)t. Solve for T and graph
the model. Compare the result with the plot of the original data.
(c) Use the graphing utility to plot the points (t, ln(T − 21)) 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 ln(T − 21) = at + b, which is equivalent to eln(T−21) = eat+b. Solve for T, 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 y-coordinates of the revised data points to generate the points (t, 1 / T - 21). 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 1 / T - 21 = at + b. Solve for T, and use the graphing utility to graph the rational function and the original data points.