A cup of water at an initial temperature of 81°C is placed in a room at a constant temperature of 24°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, 81.0°), (5, 69.0°), (10, 60.5°), (15, 54.2°), (20, 49.3°), (25, 45.4°), (30, 42.6°) (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 − 24). (b) An exponential model for the data (t, T − 24) is T − 24 = 54.4(0.964)t. Solve for T and graph the model. Compare the result with the plot of the original data. (c) Use a graphing utility to plot the points (t, ln(T − 24)) 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 − 24) = at + b, which is equivalent to eln(T − 24) = eat + b. Solve for T, and verify that the result is equivalent to the model in part (b). (Round all numerical values to three decimal places.) 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-24). Use a graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of a graphing utility to fit a line to these data. The resulting line has the form 1/T-24 = at + b. Solve for T. (Round all numerical values to four decimal places.) Use a graphing utility to graph the rational function and the original data points.
A cup of water at an initial temperature of 81°C is placed in a room at a constant temperature of 24°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, 81.0°), (5, 69.0°), (10, 60.5°), (15, 54.2°), (20, 49.3°), (25, 45.4°), (30, 42.6°)
(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 − 24).
(b) An exponential model for the data (t, T − 24) is T − 24 = 54.4(0.964)t. Solve for T and graph the model. Compare the result with the plot of the original data.
(c) Use a graphing utility to plot the points (t, ln(T − 24)) 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 − 24) = at + b, which is equivalent to eln(T − 24) = eat + b. Solve for T, and verify that the result is equivalent to the model in part (b). (Round all numerical values to three decimal places.)
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-24).
Use a graphing utility to graph these points and observe that they appear to be linear.
Use the regression feature of a graphing utility to fit a line to these data. The resulting line has the form
Solve for T. (Round all numerical values to four decimal places.)
Use a graphing utility to graph the rational function and the original data points.
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