Question #43237

A thin biconvex lens is placed in a temperature-controlled chamber. The chamber's temperature is monitored by a faulty thermometer that reads a scale $X$. The relationship between the true Celsius temperature $T_C$ and the faulty scale reading $X$ is given by $T_C = \frac{5}{9}(X - 32) + \frac{1}{90}(X - 32)^2$, where $T_C$ is in $^\circ \text{C}$ and $X$ is in $^\circ \text{X}$. At a reading of $X = 62^\circ \text{X}$, the lens has an optical power $P = +4.5 \text{ D}$, a refractive index $n = 1.6$, and a coefficient of linear thermal expansion $\alpha = 2.0 \times 10^{-5} \text{ } ^\circ \text{C}^{-1}$. The refractive index of the lens material varies with the true Celsius temperature at a rate of $\frac{dn}{dT_C} = -1.2 \times 10^{-5} \text{ } ^\circ \text{C}^{-1}$. What is the rate of change of the lens power with respect to the faulty temperature scale reading, $\frac{dP}{dX}$, at $X = 62^\circ \text{X}$?