(II) The resistance, R, of a particular thermistor as a...

(II) The resistance, $R$, of a particular thermistor as a function of temperature $T$ is shown in this Table: $$\begin{array}{cccc}\hline \boldsymbol{T}\left({ }^{\circ} \mathbf{C}\right) & \boldsymbol{R}(\boldsymbol{\Omega}) & \boldsymbol{T}\left({ }^{\circ} \mathbf{C}\right)& \boldsymbol{R}(\boldsymbol{\Omega}) \\\hline 20 & 126,740 & 36 & 60,743 \\22 & 115,190 & 38 & 55,658 \\24 & 104,800 & 40 & 51,048 \\26 &95,447 & 42 & 46,863 \\28 & 87,022 & 44 & 43,602 \\30 & 79,422 & 46 & 39,605 \\32 & 72,560 & 48 & 36,458 \\34 & 66,356 & 50 & 33,591 \\\hline\end{array}$$ Determine what type of best-fit equation (linear, quadratic, exponential, other) describes the variation of $R$ with $T$. The resistance of the thermistor is $57,641 \Omega$ when embedded in a substance whose temperature is unknown. Based on your equation, what is the unknown temperature?

(II) The resistance, $R$, of a particular thermistor as a function of temperature $T$ is shown in this Table:
$$\begin{array}{cccc}\hline \boldsymbol{T}\left({ }^{\circ} \mathbf{C}\right) & \boldsymbol{R}(\boldsymbol{\Omega}) & \boldsymbol{T}\left({ }^{\circ} \mathbf{C}\right)& \boldsymbol{R}(\boldsymbol{\Omega}) \\\hline 20 & 126,740 & 36 & 60,743 \\22 & 115,190 & 38 & 55,658 \\24 & 104,800 & 40 & 51,048 \\26 &95,447 & 42 & 46,863 \\28 & 87,022 & 44 & 43,602 \\30 & 79,422 & 46 & 39,605 \\32 & 72,560 & 48 & 36,458 \\34 & 66,356 & 50 & 33,591 \\\hline\end{array}$$
Determine what type of best-fit equation (linear, quadratic, exponential, other) describes the variation of $R$ with $T$. The resistance of the thermistor is $57,641 \Omega$ when embedded in a substance whose temperature is unknown. Based on your equation, what is the unknown temperature?

Key Concepts

Linearization Techniques

When dealing with exponential relationships, linearization techniques—such as taking the natural logarithm of the dependent variable—can simplify the analysis. By transforming an exponential equation into a linear one, it becomes easier to use standard linear regression methods to determine the coefficients, effectively redefining the model in a form that is simpler to analyze and interpret.

Function Inversion for Solution

Once an appropriate model is established, inverting the function becomes necessary when solving for an independent variable given a particular dependent variable value. In the context of an exponential model, this typically involves taking logarithms to isolate the temperature variable when a specific resistance value is provided. This inversion is a common step in many physical applications where the measurement of one quantity is used to infer another.

Exponential Function Modeling

Many physical phenomena, including the behavior of thermistors, are best described by exponential functions. In such models, a variable changes at a rate proportional to its current value, resulting in a curve that decreases (or increases) rapidly at first and then levels off. Recognizing that the relationship between resistance and temperature follows this pattern can lead to an exponential model rather than a simple linear or quadratic one.

Curve Fitting Analysis

Curve fitting is a statistical tool used to determine the mathematical relationship between measured data points. In cases like resistance versus temperature, different types of fits (linear, quadratic, exponential, etc.) are tested to see which model best describes the observed trend. Determining that the exponential model provides the best fit requires comparing the residuals or errors from each candidate model.

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