Chapter 3.6, Problem 57E

Temperature distribution A thin copper rod, 4 meters in length, is heated at its midpoint, and the ends are held at a constant temperature of 0°. When the temperature reaches equilibrium, the temperature profile is given by T(x) = 40x(4 − x), where 0 ≤ x ≤ 4 is the position along the rod. The heat flux at a point on the rod equals −kT′(x), where k > 0 is a constant. If the heat flux is positive at a point, heat moves in the positive x-direction at that point, and if the heat flux is negative, heat moves in the negative x-direction.

1. a. With k = 1, what is the heat flux at x = 1? At x = 3?
2. b. For what values of r is the heat flux negative? Positive?
3. c. Explain the statement that heat flows out of the rod at its ends.