Chapter 8.2, Problem 25P

Show that the thickness of the ice on a lake increases with the square root of the time in cold weather, making the following simplifying assumptions. Let the water temperature be a constant ${10}^{°}\text{C},$ the air temperature a constant $-{10}^{°},$ and assume that at any given time the ice forms a slab of uniform thickness x. The rate of formation of ice is proportional to the rate at which heat is transferred from the water to the air. Let $t=0$ when $x=0$.