Question

The initial temperature distribution along the length of the rod is given above. Final temperature distribution after a long time, can be expressed as
\left( A \right)T = 75\cos {\left( {15.7x} \right)^ \circ }C \\\ \left( B \right)T = {25^ \circ }C \\\ \left( C \right)T = 25\tan {\left( {15.7x} \right)^ \circ }C \\\ \left( D \right)T = {0^ \circ }C \\\

Answer 1

Hint : In order to solve the question, we will refer to the temperature distribution relation inside a rod that refers to the variation of temperature along with the length of the rod and that of heat along with the time. Generally the temperature first falls off along the length and then increases.
The formula used here is the temperature distribution formula along with length
$\dfrac{{dQ}}{{dt}} = k\dfrac{{dT}}{{dx}}$
Where k is some constant of proportionality
Here, $\dfrac{{dQ}}{{dt}}$ is rate of change of heat with respect to time
$\dfrac{{dT}}{{dx}}$ is the rate of change of temperature with respect to the distance within the rod.

Complete Step By Step Answer:
Inside a rod, there is a continuous heat flow from the rod along the length. The relation of loss of heat along the length of the rod with the change of temperature is given by equation
$\dfrac{{dQ}}{{dt}}\alpha \dfrac{{dT}}{{dx}}$
This means that there is a continuous loss of heat throughout the rod, therefore, the temperature increases as we move along the length till the middle and then will start to decrease. Hence, we will reach a point where the rod will be uniform in this distribution. That point is achieved when the temperature of the rod after decreasing reaches to zero degrees Celsius. Therefore,
At $T = {0^ \circ }C$ , the rod will be uniform.
Among the above four options, the right answer is given by option $\left( D \right)$

Note :
The rate of change of heat with respect to the time inside the rod is directly proportional to the rate of change of temperature with respect to the distance within the rod. Along the length, the temperature first increases and then decreases. This gives a uniformity at zero degrees celsius temperature.