This is a mathematical modeling question on optimizing the icing on a cake. A we

Question

This is a mathematical modeling question on optimizing the icing on a cake. A wedding cake is to be baked in a circular cake tin and will have a volume of 3,977 cm^3. Let the radius of the cake be r cm and the height of the cake be h cm. Determine the height h, in cm, of the cake which will give minimum surface area for the icing (i.e. the top area and the curved surface area at the side). Give your answer to 3 decimal places, and use pi = 3.142. Find expressions for volume V, and surface area S, of the cake in terms of rand h. Then differentiate S with respect to r, and then use calculus principles to find rat which Sis minimum.

Explanation / Answer

V = (pi)hr^2 so h = V/((pi)r^2)
S = 2(pi)rh + (pi)r^2
Therefore by substitution:
S = 2V/r + (pi)r^2 = 7954/r + 3.142r^2

Min S when dS/dr = 0 and d^2S/dr^2 > 0

dS/dr = 6.284r - 7954/r^2
dS/dr = 0 when r = 10.816 cm .

h = 10.816 cm..................Ans.

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