b. Explain why the data point, temperature 35 °C at 270 seconds, should not be used in the linear fit of the high temperature data.
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Step 1: The question asks to explain why a specific data point (35 °C at 270 seconds) should not be used in a linear fit of *high temperature data*. Show more…
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The following table gives the temperature, in degrees Celsius, of a cup of hot water sitting in a room with constant temperature. The data was collected over a period of 30 minutes. (Source: www.phys. unt.edu, Dr. James A. Roberts)$$\begin{array}{|c|c|} \hline\text { Time } & \text { Temperature } \\(\mathrm{min}) & (\text { degrees Celsius }) \\ \hline0 & 95 \\1 & 90.4 \\5 & 84.6 \\10 & 73 \\15 & 64.7 \\20 & 59 \\25 & 54.5 \\29 & 51.4\\\hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form $f(t)=C a^{2}$ that best fits the data. Let $t$ be the number of minutes the water has been cooling. (b) Using your modicl, what is the projected temperature of the water after 1 hour?
Exponential and Logarithmic Functions
Exponential, Logistic, and Logarithmic Models
A beaker of liquid at an initial temperature of $78^{\circ} \mathrm{C}$ is placed in a room at a constant temperature of $21^{\circ} \mathrm{C}$ The temperature of the liquid is measured every 5 minutes for a period of $\frac{1}{2}$ hour. The results are recorded in the table, where $t$ is the time (in minutes) and $T$ is the temperature (in degrees Celsius). $$\begin{array}{|c|c|}\hline \text { Time, } t 0 & \text { Temperature, } T\\\hline 0 & 78.0^{\circ} \\5 & 66.0^{\circ} \\10 & 57.5^{\circ} \\15 & 51.2^{\circ} \\20 & 46.3^{\circ} \\25 & 42.5^{\circ} \\30 & 39.6^{\circ} \\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a linear model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window. Do the data appear linear? Explain. (b) Use the regression feature of the graphing utility to find a quadratic model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window. Do the data appear quadratic? Even though the quadratic model appears to be a good fit, explain why it might not be a good model for predicting the temperature of the liquid when $t=60.$ (c) The graph of the temperature of the room should be an asymptote of the graph of the model. Subtract the room temperature from each of the temperatures in the table. Use the regression feature of the graphing utility to find an exponential model for the revised data. Add the room temperature to this model. Use the graphing utility to plot the original data and graph the model in the same viewing window. (d) Explain why the procedure in part (c) was necessary for finding the exponential model.
Nonlinear Models
Q5. The failure rate of certain electronic device is suspected to increase linearly with its temperature. Fit a least-squares linear line through the data in the following table. Table: The Failure Rate versus Temperature Temp: 55, 65, 75, 85, 95, 105, 55, 65, 75, 85, 95, 105 Failure Rate: 1.90, 1.93, 1.97, 2.00, 2.01, 2.01, 1.94, 1.95, 1.97, 2.02, 2.02, 2.04 a. Draw a scatter diagram. Comment on it. b. Estimate a simple linear regression equation and interpret the equation. c. Determine correlation coefficient and coefficient of determination and comment on them.
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