In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line.

One may conclude that

Step-by-Step Solution

1. Identify the Equation of a Straight Line

From the statement, the graph plotted is a straight line. The standard equation for a downward-sloping straight line with a positive vertical intercept is given by:

$$y = -mx + c$$

Where:

• $$y$$ represents the variable on the vertical axis.
• $$x$$ represents the variable on the horizontal axis.
• $$m$$ is the positive slope magnitude.
• $$c$$ is the positive vertical intercept.
• The vertical axis variable ($$y$$) is $$\ln(R)$$.
• The horizontal axis variable ($$x$$) is $$\frac{1}{T^2}$$.
• $$R_0 = e^c$$
• $$m = T_0^2$$

2. Match Axis Variables to the Graph

By looking at the provided mathematical steps:

Substituting these coordinates into the straight-line equation:

$$\ln(R) = -m\left(\frac{1}{T^2}\right) + c$$

3. Convert logarithmic form to Exponential form

To find the direct relationship for resistance $$R$$ as a function of temperature $$T$$, take the exponential base $$e$$ on both sides of the equation:

$$e^{\ln(R)} = e^{-\frac{m}{T^2} + c}$$

Using exponential properties ($$e^{a+b} = e^a \cdot e^b$$):

$$R = e^c \cdot e^{-\frac{m}{T^2}}$$

Let the constant term $$e^c$$ be defined as the baseline resistance $$R_0$$, and the slope constant $$m$$ be defined as a characteristic temperature squared parameter $$T_0^2$$:

Substituting these constants back gives the function:

$$R(T) = R_0 e^{-\frac{T_0^2}{T^2}}$$

Final Answer

The variation of resistance $$R$$ with temperature $$T$$ that satisfies the linear graph is given by the exponential relationship:

$$R = R_0 e^{-\frac{T_0^2}{T^2}}$$