Some young entrepreneurs started an industry “young achievers” for casting metal into various shapes. They put up an advertisement online stating the same and expecting order to cast metal for toys sculptures, decorative pieces and more. A group of friends wanted to make innovative toys and hence contacted the “young achievers” to order them to cast metal into solid half cylinders with a rectangular base and semi-circular ends.

Based on the above information, answer the following questions:
For the given half-cylinder of volume V, the total surface area S is minimum, when:

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$(π + 2) V = π^{2} r^{2}$

$2 (π + 2) V = π^{2} r^{3}$

$(π + 2) V = π^{2} r^{3}$

$(π + 2) V = π^{2} r$

answer is A.

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Detailed Solution

$\begin{aligned} S = π r^{2} + \frac{2 V (π + 2)}{π r} \\ \Rightarrow \frac{d S}{d r} = 2 π r - \frac{2 V (π + 2)}{π r^{2}} \end{aligned} F o r m a x o r m i n \frac{d S}{d r} = 0 \Rightarrow 2 π r = \frac{2 V (π + 2)}{π r^{2}} \Rightarrow π^{2} r^{3} = (π + 2) V$