The resistance of a resistance thermometer has values \(2.71\) and $3.70 \mathrm{ohm}\( at \)10^{\circ} \mathrm{C}\( and \)100^{\circ} \mathrm{C}$. The temperature at which the resistance is \(3.26 \mathrm{ohm}\) is (A) \(40^{\circ} \mathrm{C}\) (B) \(50^{\circ} \mathrm{C}\) (C) \(60^{\circ} \mathrm{C}\) (D) \(70^{\circ} \mathrm{C}\)

First, we will use the given data points to find the slope between temperature and resistance. The slope can be calculated using the formula: Slope = \(\frac{Change\ in\ Resistance}{Change\ in\ Temperature}\) So, Slope = \(\frac{3.70 \mathrm{ohm} - 2.71 \mathrm{ohm}}{100^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C}}\)

Now, let's calculate the slope using the given values: Slope = \(\frac{3.70 \mathrm{ohm} - 2.71 \mathrm{ohm}}{100^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C}}\) = \(\frac{0.99 \mathrm{ohm}}{90^{\circ} \mathrm{C}}\) = \(0.011 \mathrm{ohm}/^{\circ}\mathrm{C}\)

With the slope value, we can now find the linear equation that relates temperature (T) and resistance (R). Using one of the given data points (let's use \(10^{\circ}\mathrm{C}\) and \(2.71 \mathrm{ohm}\)): \(R - R_1 = Slope \times (T - T_1)\) where \(R_1\) is the known resistance value, and \(T_1\) is the known temperature value. Now, let's plug in the values. We have: \(R - 2.71 \mathrm{ohm} = 0.011 \frac{\mathrm{ohm}}{^{\circ}\mathrm{C}} \times (T - 10^{\circ}\mathrm{C})\)

We are given the resistance value as \(3.26\mathrm{ohm}\). Now, we will plug this into our equation and solve for the temperature (T): \(3.26 \mathrm{ohm} - 2.71 \mathrm{ohm} = 0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}} \times (T - 10^{\circ}\mathrm{C})\) Next, we will simplify and solve for T: \(0.55 \mathrm{ohm} = 0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}} \times (T - 10^{\circ}\mathrm{C})\) To solve for T, we will divide both sides by the slope: \(\frac{0.55\mathrm{ohm}}{0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}}} = T - 10^{\circ}\mathrm{C}\) Now, we can calculate the value of T: \(T = 10^{\circ}\mathrm{C} + \frac{0.55\mathrm{ohm}}{0.011\frac{\mathrm{ohm}}{^{\circ}\mathrm{C}}} = 10^{\circ}\mathrm{C} + 50^{\circ}\mathrm{C} = 60^{\circ}\mathrm{C}\) So, the temperature at which the resistance is \(3.26 \mathrm{ohm}\) is \(60^{\circ}\mathrm{C}\), which corresponds to option (C).