A thermostat control with dial markings from 0 to 100 is used to regulate the temperature of an oil bath. A calibration plot on logarithmic coordinates of the temperature, T(°F), versus the dial setting, R, is a straight line that passes through the points (R₁ = 20.0; T₁ = 110.0°F) and (R₂ = 40.0.; T₂ = 250.0°F). 

(a) Derive an equation for T(°F) in terms of R. 

(b) Estimate the thermostat setting needed to obtain a temperature of 320°F. 

(c) Suppose you set the thermostat to the value of R calculated in Part (b), and the reading of a thermocouple mounted in the bath equilibrates at 295°F instead of 320°F. Suggest several possible explanations.