Question

The variation of temperature in a plane wall is determined to be $T(x)=110 x-48 x\( where \)x\( is in \)\mathrm{m}\( and \)T\( is in \){ }^{\circ} \mathrm{C}$. If the thickness of the wall is \(0.75 \mathrm{~m}\), the temperature difference between the inner and outer surfaces of the wall is (a) \(110^{\circ} \mathrm{C}\) (b) \(74^{\circ} \mathrm{C}\) (c) \(55^{\circ} \mathrm{C}\) (d) \(36^{\circ} \mathrm{C}\) (e) \(18^{\circ} \mathrm{C}\)

Short answer

Answer: The temperature difference between the inner and outer surfaces of the wall is \(74^{\circ} \mathrm{C}\).

Step-by-step solution

Calculate temperature at inner surface (x=0)

To calculate the temperature at the inner surface, we have to plug x = 0 in the given function: \(T(0) = 110(0) - 48(0)^2 = 0^{\circ} \mathrm{C}\)

Calculate temperature at outer surface (x=0.75)

To calculate the temperature at the outer surface, we have to plug x = 0.75 into the given function: \(T(0.75) = 110(0.75) - 48(0.75)^2= 74^{\circ} \mathrm{C}\)

Calculate the temperature difference

Now we can calculate the temperature difference between the inner and outer surfaces by subtracting inner surface temperature from the outer surface temperature: \(\Delta T = T(0.75) - T(0) = 74^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C} = 74^{\circ} \mathrm{C}\) The temperature difference between the inner and outer surfaces of the wall is \(74^{\circ} \mathrm{C}\), so the correct answer is (b).