This is a mathematical modelling question on optimizing the icing on a cake. A w

Question

This is a mathematical modelling 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 3,907 cm3. 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 ? = 3.142.

Hint : Find expressions for volume V, and surface area S, of the cake in terms of r and h. Then differentiate S with respect to r, and then use calculus principles to find r at which S is 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 = 7814/r + 3.142r^2

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

dS/dr = 6.284r - 7814/r^2
dS/dr = 0 when r = 12.44 cm

h = 12.44 cm (3 decimals)

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