(Harvey's potato problem) Take a (theoretical) potato, slice it into slices of thickness L, and drop the slices into a large pot of boiling water. If L is much smaller than the diameter of the potato, then almost all of the heat entering a slice of potato comes through its two cut surfaces, and we can use a one-dimensional model to describe the cooking process, at least approximately. If the potato began at room temperature, and if the slices are not too close to each other in the boiling water, then the temperature, u(x, t), of a potato slice of thickness L > 0 can be modeled by: PDE: au = k t x2 BC: u(0,t) = 100C, u(L,t) = 100C, IC: u(x, 0) = 20C, 0 0, 0 < x < L. Fun Fact: The thermal diffusivity of potato is k = 0.0017cm/sec. (see R. P. SINGH, Food Technology, vol. 36, pp. 8791, 1982.) (a) Let u(x) be the solution to the steady-state problem and let v(x, t) be the solution of the transient problem and suppose u(x,t) = u(x) + v(x,t). Find the steady-state solution, u(x). (b) Write down explicitly the set of equations that define the transient solution v(x, t) and show how you arrived at them. (c) Write down the full solution, u(x,t) = u(x) + v(x,t), of the problem. (For this part, you do not need to show all the steps of separation of variables when solving for v(x, t).) (d) Define "cooking" to be a process in which we raise the temperature of the object enough to break down its cell walls. The potato slice is cooked when the temperature of the center of the slice is 75C. If the potato is cut into slices 1 cm thick (so L = 1), approximately how long does it take a slice to cook? (Use the first Fourier mode of the temperature at the center of the slice to approximate the time.) (e) In fact, the boiling temperature of water is 100C only at sea level. In Boulder, the air pressure is lower, and water boils at about 94C. How long does it take to cook a 1 cm thick slice of potato in Boulder? Justify. [If you moved to Boulder from a lower elevation and tried to bake a cake, you may have noticed the secret family recipe needs some adjustment here.] (f) Suppose you deep-fried the potato (in oil) instead of boiling it (in water). Now the temperature at the boundaries is 200C instead of 100C. What is the cooking time now? Justify.