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A given quantity of metal is to be cast into a half-cylinder, which has a rectangular base and semicircular cross-section at both ends. For minimum total surface area, the ratio of the length (height) of the half-cylinder to the diameter of the semicircular ends equals pi / (pi + k). Find the value of k.

1. 2
2. 3
3. 4
4. 6

Correct answer: 4

Solution

Set up: Volume = (pi*r²/2)*l = const => l = 2V/(pi*r²). Surface area S = rectangular base (2r*l) + two semicircular ends (pi*r²/2 + pi*r²/2 = pi*r²) + curved top (pi*r*l). So S = 2rl + pi*r² + pi*r*l = l*r*(2 + pi) + pi*r². Substituting l: dS/dr = 0 gives ratio l/d = l/(2r) = pi/(pi+4), so k = 4.

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