Rework Exercise 13 for a metal with a constant of proportionality k = .2. Which rod cools faster, the rod with a constant of proportionality k = .1 or the rod with a constant of proportionality k = .2? What can you say about the effect of varying the constant of proportionality in a cooling problem?

Exercise 13

When a red-hot steel rod is plunged in a bath of water that is kept at a constant temperature 10C, the temperature of the rod at time t, f (t), satisfies the differential equation y′ = k[10 - y], where k > 0 is a constant of proportionality. Determine f (t) if the initial temperature of the rod is f (0) = 350°C and k = .1.