10.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-22_451_506_178_767} The figure above shows solid right prism of height \(h \mathrm {~cm}\). The cross section of the prism is a circular sector of radius \(r \mathrm {~cm}\), subtending an angle of 2 radians at the centre.
a) Given that the volume of the prism is \(1000 \mathrm {~cm} ^ { 3 }\), show clearly that $$S = 2 r ^ { 2 } + \frac { 4000 } { r } ,$$ where \(S \mathrm {~cm} ^ { 2 }\) is the total surface area of the prism.
b) Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer.
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