Chapter 12, Problem 12PS

a.

To determine

Use the regression feature of a graphing utility to find a model of the form

  $T_1=a{t}^2+bt+c$ for the data.

Answer to Problem 12PS

  $\boxed{y=-0.003x^2+0.677x+26.563}$

Explanation of Solution

Given information:

The heat exchanger of a heating system has a heat probe attached to it. The temperature $T$ (in degrees Celsius) is recorded $t$ seconds after starting the furnace. The table shows the results for the first $2$ minutes.

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Use the regression feature of a graphing utility to find a model of the form

  $T_1=a{t}^2+bt+c$ for the data.

Calculation:

Consider the following data

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

To find the quadratic regression model using the graphing utility $Ti-83$ , proceed as follows.

Press stat. The display will be

  

Now press edit. The display will be

  

Now type in data provided. The display will be

  

Now press stat and choose calc. The display will be

  

Now press QuadReg. The display will be

  

Hence, the quadratic regression model of the baove data is

  $\boxed{y=-0.003x^2+0.677x+26.563}$

b.

To determine

Verify that the model fit according to the data.

Answer to Problem 12PS

The model fits the data well with least devition.

Explanation of Solution

Given information:

The heat exchanger of a heating system has a heat probe attached to it. The temperature $T$ (in degrees Celsius) is recorded $t$ seconds after starting the furnace. The table shows the results for the first $2$ minutes.

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Use the graphing utility to graph $T_1$ with the original data. How well does the model fit the data?

Calculation:

Consider the following data

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

To plot the function, proceed as follows.

Press 2^nd$Y$ and select $1$. The display will be

  

Now select on under plot $1$ . The display will be

  

Now choose window and choose the proper scale. The display will be

  

Now choose $Y$ and type in the function obtained in the quadreg mode. The display will be

  

Now press graph. The display will be

  

The derivation between the scatter plot of the points and the equation obtained from regression model is minimal.

Hence, the model fits the data well with least devition.

c.

To determine

Verify that the model fit according to the data.

Answer to Problem 12PS

The model fits the data well with least deviation.

Explanation of Solution

Given information:

The heat exchanger of a heating system has a heat probe attached to it. The temperature $T$ (in degrees Celsius) is recorded $t$ seconds after starting the furnace. The table shows the results for the first $2$ minutes.

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

A rational model for the data is given by

  $T_2=\frac{86t+1451}{t+58}$

Use the graphing utility to graph $T_2$ with the original data. How well does the model fit the data?

Calculation:

Consider the following data

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Consider the rational model of the data

  $T_2=\frac{86t+1451}{t+58}$

To graph the model using the graphing utility with original values of $t$ , proceed as follows .

Press stat and choose edit. The display will be

  

Now type in the function and display will be

  

Now choose 2^nd$Y$ and choose plot 2. Change $Y$ list to L3. The display will be

  

Now press graph. The display will be

  

The deviation between the scatter plot of the points, the equation obtained from the regression model and the data evaluated for $T_2$ is minimal.

Hence, the model fits the data well with least deviation.

d.

To determine

Evaluate $T_1(0)$ and $T_2(0)$ .

Answer to Problem 12PS

  $\boxed{T_2(0)=25.012°}$

Explanation of Solution

Given information:

The heat exchanger of a heating system has a heat probe attached to it. The temperature $T$ (in degrees Celsius) is recorded $t$ seconds after starting the furnace. The table shows the results for the first $2$ minutes.

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Evaluate $T_1(0)$ and $T_2(0)$ .

Calculation:

Consider the following data

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Now move up the arrow key to obtain the value of $T_2(0)$ . The display will be

  

Hence, the value $T_2(0)$ is $\boxed{T_2(0)=25.012°}$

e.

To determine

Find the $\underset{t\to \infty }{\lim }{T}_2$ V

Answer to Problem 12PS

  $\boxed{86°}$

Explanation of Solution

Given information:

The heat exchanger of a heating system has a heat probe attached to it. The temperature $T$ (in degrees Celsius) is recorded $t$ seconds after starting the furnace. The table shows the results for the first $2$ minutes.

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Find $\underset{t\to \infty }{\lim }{T}_2$ Verify your result using the graph in part (c).

Calculation:

Consider the following data

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Now consider the limit of the function

  $\underset{t\to \infty }{\lim }{T}_2$

To determine the limit of the function, proceed as follows.

  $\underset{t\to \infty }{\lim }{T}_2=\underset{t\to \infty }{\lim }\frac{86t+1451}{t+58}$

Divide both numerator and denominator by $t$

  $\begin{array}{l}\underset{t\to \infty }{\lim }{T}_2=\underset{t\to \infty }{\lim }\frac{86+\frac{1451}{t}}{1+\frac{58}{t}}\\ =\frac{86}{1}\end{array}$

  $\boxed{86°}$

f.

To determine

Explain your reasoning.

Answer to Problem 12PS

The functional relationship between temperature and time is discrete in nature.

Explanation of Solution

Given information:

The heat exchanger of a heating system has a heat probe attached to it. The temperature $T$ (in degrees Celsius) is recorded $t$ seconds after starting the furnace. The table shows the results for the first $2$ minutes.

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

Interpret the result of part (e) in the context of the problem. Is it possible to perform this type of analysis using $T_1$ ? Explain your reasoning.

Calculation:

Consider the following data

  $\begin{array}{cc}t& T\\ 0& 25.2°\\ 15& 36.9°\\ 30& 45.5°\\ 45& 51.4°\\ 60& 56.0°\\ 75& 59.6°\\ 90& 62.0°\\ 105& 64.0°\\ 120& 65.2°\end{array}$

The time $t$ approaches infinity the temperature of the heat system according to the rational model is $86C$ .

This type of analysis is not possible using the data provided for $T_1$ ,as the data is a raw data.

Hence, the functional relationship between temperature and time is discrete in nature.