A student connects a real battery of unknown electromotive force \(\mathcal{E}\) and internal resistance \(r\) to a variable resistor of calibrated resistance \(R\). An ideal voltmeter connected across the battery terminals measures the terminal voltage \(\Delta V\) for several different values of \(R\). Which of the following pairs of quantities should be plotted on the vertical and horizontal axes to produce a linear graph, and what is the correct expression for \(r\) in terms of the slope \(m\) and vertical intercept \(b\) of the resulting best-fit line?