Let height and radius of a right circular cylinder are \( h \) and \( r \) respectively. If it is open at the top, having a given surface area and greatest volume, then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( h=r \)
A cylinder open at the top has one circular base and the curved surface. Let the given surface area be S.
\( \displaystyle S=\pi r^2+2\pi rh \)
\( \displaystyle h=\frac{S-\pi r^2}{2\pi r} \)
\( \displaystyle V=\pi r^2h \)
\( \displaystyle =\pi r^2\cdot\frac{S-\pi r^2}{2\pi r} \)
\( \displaystyle =\frac{Sr-\pi r^3}{2} \)
\( \displaystyle \frac{dV}{dr}=\frac{S-3\pi r^2}{2}=0 \)
\( S=3\pi r^2 \)
\( \displaystyle \frac{d^2V}{dr^2}=-3\pi r\lt 0, \text{ so V is greatest} \)
Putting S = 3πr2 back into the surface area:
\( 3\pi r^2=\pi r^2+2\pi rh \)
\( 2\pi r^2=2\pi rh \)
\( h=r \)
Why the other options are incorrect
Option (1) \( 2h=r \) — Incorrect. \( 2h=r \) does not satisfy \( S=3\pi r^2 \): it gives \( S=\pi r^2+\pi r^2=2\pi r^2 \).
Option (2) \( h=2r \) — Incorrect. h = 2r is the condition for the greatest volume of a closed cylinder of given total surface area (two circular ends).
Option (4) \( h=\sqrt{2}r \) — Incorrect. With \( h=\sqrt{2}r \), \( S=\pi r^2+2\sqrt{2}\pi r^2\ne 3\pi r^2 \).