Partial Differential Equations (Dirichlet Boundary Condition; Separation of Variables)
Explain in some detail, the reason for your answer:
Consider a thin rod of heat-conducting material with length L. Suppose that the rod is initially heated to a temperature of T uniformly throughout the tod, and is dropped into a bucket of ice water at t = 0. Suppose that the rod is everywhere insulated, except for its left end (x = 0), which is expoosed to the ice water.
(a) Write down the system of equations that describes this scenario. (You can use Dirichlet boundary condition for the left end of the rod.)
(b) Find the temperature of the rod using seperation of variables.
(c) At any given time, which part of the rod will be the warmest? Estimate when the maximum temperature of the rod will be less than 1% of T.