GATE (TF) Textile 2010 Question Paper Solution | GATE/2010/TF/47

For a circular rod with volume $16\pi cm^3$ the value of radius for which the surface area (including the top and bottom surfaces) will be minimum is

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Volume= $16\pi cm^3$

As we know,

Volume of cylinder= $\pi\times r^2 \times h$

Where, r=radius and h is the height

$\pi\times r^2 \times h=16\pi$

$r^2\times h=16$

$h=\frac{16}{r^2}$

When surface area is minimum-

Surface area(S)= $2\pi\times r \times h+2\pi\times r^2$

S= $2\pi\times r (r+h)$

S= $2\pi\times r (r+\frac{16}{r^2})$

S= $2\pi (r^2+\frac{16}{r})$

Differentiate this with respect to r-

$\frac{dS}{dr}=2\pi (2r-\frac{16}{r^2})$ =0

$\frac{16}{r^2}=2r$

$r^3=8$

r=2 cm (Ans)