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Question

A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum.

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Solution

Let the height, radius of the base and surface area of the cylinder be h, r and S, respectively. Then,Volume =πr2h2156=πr2h2156=227r2hh=2156×722r2h=686r2 ...1Surface area=2πrh+2πr2S=4312r+44r27 From eq. 1dSdr=4312-r2+88r7For maximum or minimum values of S, we must havedSdr=04312-r2+88r7=04312r2=88r7r3=4312×788r3=343r=7 cmNow, d2sdr2=8624r3+887d2sdr2=8624343+887d2sdr2=1767>0So, the surface area is minimum when r=7 cm.

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