When baking potatoes, the main limiting factor is getting the temperature of the entire potato up to 100C. By cutting a potato into smaller pieces, you can deliver heat more rapidly.Although the analysis described here can be carried out with similar conclusions for spherical and even ellipsoidal potatoes, given the constraints of the course content, we will consider cube-like chunks of potato that areperfectlyinsulatedonfoursidesandexposedtoanoventemperatureof190Conthetopand bottomsurface.Thisgeometryleadstouniform temperaturesinthe x and y directionsandonly variation in the z direction.The temperature of the potato u(z,t) at height z changes according to the Heat Equation. ut=Duzz Initially, the temperature of the potato is 20C everywhere from bottom (z= 0) to top (z=L).The oven is kept at a constant temperature 190C so that the bottom and top surfaces of the potato chunk are always 190C.

• Express the initial condition in the usual mathematical format.
• Express the boundary conditions (BCs) in the usual mathematical format.
• Find the steady state solution to the Heat Equation that solves the BCs.
• Express the general solution to the PDE in terms of the steady state and correct type of Fourier series (leaving the arbitrary constant unspecified for now).
• Find values for the arbitrary coefficients in your general solution to get the particular solution for the IC. Identify any patterns of zeros in the terms in the Fourier series and make these explicit in how you express your bn.Your solution should not include terms that have a zero coefficient.
• The temperature at all points in the potato go from the initial 20C to 190C as t .The part of the potato chunk at x=L/2 is the slowest to heat up.Write down the temperature at the middle of the chunk,u(L/2,t).
• Which of the modes (Fourier terms) decays slowest?Ignore the others and use that slowest modeto find an expression for the time at whichu(L/2,t) = 100.
• You initially cut the chunks into pieces of size L= 4 cm.From experience, you know it will take 40minutesforthesetocook.Butyou'reinarush.Isitworthtakinganother20minutesto cut them into pieces of size L= 2 cm?A naive assumption might be that pieces half the size would take half the time to cook so it would not be any faster to do the extra cutting (40 minutes cooking versus 20 minutes cutting + 20 minutes cooking).Use your answer to the previous part to estimate the cooking time more accurately and make a more reliable decision.Justify your answer.