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Chocolate covered candy

Problem 449

Published on Sunday, 8th December 2013, 04:00 am; Solved by 340

Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: b2x2 + b2y2 + a2z2 = a2b2.

Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick.

If a=1 mm and b=1 mm, the amount of chocolate required is
28
3
π mm3
If a=2 mm and b=1 mm, the amount of chocolate required is approximately 60.35475635 mm3.

Find the amount of chocolate in mm3 required if a=3 mm and b=1 mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.