A cylinder with height ‘\( h \)’ and radius of base is ‘\( r \)’ which is open at the top having a given outer surface area , then volume of the cylinder is maximum if
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( h=r \)
An open cylinder has a base and a curved surface, but no top. Let the fixed surface area be S.
\( S=\pi r^2+2\pi rh \)
\( \displaystyle h=\frac{S-\pi r^2}{2\pi r} \)
\( \displaystyle V=\pi r^2h=\frac{Sr-\pi r^3}{2} \)
\( \displaystyle \frac{dV}{dr}=\frac{S-3\pi r^2}{2} \)
Setting \( \frac{dV}{dr}=0 \) gives \( S=3\pi r^2 \). Also \( \frac{d^2V}{dr^2}=-3\pi r\lt 0 \), so V is maximum. Then
\( 2\pi rh=S-\pi r^2=2\pi r^2 \)
\( h=r \)
Why the other options are incorrect
Option (1) \( r=2h \) — Incorrect. The maximum needs h = r, not r = 2h.
Option (2) \( h=3r \) — Incorrect. The maximum needs h = r, not h = 3r.
Option (4) \( h=2r \) — Incorrect. This is the condition for a closed cylinder; here the top is open.