Chapter 3.3, Problem 90E

Writing Write a short paragraph explaining why the transformations of the data in Exercise 89 were necessary to obtain the models. 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?

89. Comparing Models A cup of water at an initial temperature of ${78}^{\circ }C$ is placed in a room at a constant temperature of ${21}^{\circ }C$. 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).

$\left(0,{78.0}^{\circ }\right),\left(5,{66.0}^{\circ }\right),\left(10,{57.5}^{\circ }\right),\left(15,{51.2}^{\circ }\right),\left(20,{46.3}^{\circ }\right),\left(25,{42.4}^{\circ }\right),\left(30,{39.6}^{\circ }\right)$

(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 $\left(t,\ln \left(T-21\right)\right)$ 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 $e^{\ln (T-21)}=e^{at+b}$. 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.