Chapter 5.3, Problem 89E

Comparing Models 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 $\left(t,T\right),$ where $t$ is the time (in minutes) and $T$ is the temperature (in degrees Celsius).

$\begin{array}{l}\left(0,78.0°\right),\left(5,66.0°\right),\left(10,57.5°\right),\left(15,51.2°\right),\hfill \\ \left(20,46.3°\right),\left(25,42.4°\right),\left(30,39.6°\right)\hfill \end{array}$

(a) Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points $\left(t,T\right),$ and $\left(t,T-21\right),$

(b) An exponential model for the data $\left(t,T-21\right),$ is $T-21=54.4{\left(0.964\right)}^t$ 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 $\left(t,1\text{n}\left(T-21\right)\right)$ 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 $\ln \left(T-21\right)=at+b$, which is equivalent to $e^{\ln \left(T-21\right)}=e^{at+b}$ 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

$\left(t,\frac{1}{T-21}\right).$

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

$\frac{1}{T-21}=at+b$

Solve for $T$, and use the graphing utility to graph the rational function and the original data points.