A closed right circular cylinder has a volume of 2156 cm3. The radius…

The correct answer is Option (1) → 7 cm

Let the radius be r and the height h. The volume fixes h:

    \( \displaystyle \pi r^2h=2156 \)

    \( \displaystyle h=\frac{2156}{\pi r^2} \)

The total surface area of a closed cylinder is

    \( \displaystyle S=2\pi r^2+2\pi rh \)

    \( \displaystyle =2\pi r^2+\frac{4312}{r} \)

    \( \displaystyle \frac{dS}{dr}=4\pi r-\frac{4312}{r^2}=0 \)

    \( \displaystyle r^3=\frac{4312}{4\pi}=\frac{1078}{\pi} \)

Taking \( \pi=\frac{22}{7} \):

    \( \displaystyle r^3=\frac{1078\times 7}{22}=343 \)

    \( \displaystyle r=7\ \text{cm} \)

    \( \displaystyle \frac{d^2S}{dr^2}=4\pi+\frac{8624}{r^3}\gt 0 \)

So S is minimum at r = 7 cm.

Tip: For a closed cylinder of fixed volume, the surface area is least when the height equals the diameter (h = 2r); here h = 14 = 2 × 7.