Question

If an open cylinder of given surface area has maximum volume, then its radius is

• A. Height of the cylinder
• B. Height of the cylinder \(/2\)
• C. \(2\) times Height of the cylinder
• D. \(3\) times Height of the cylinder

Hint:

For optimization problems involving cylinders, first express the volume in terms of a single variable using the surface area constraint, then differentiate and apply the second derivative test.

Answer 1

The Correct Option is A

Step 1: Express volume in terms of r.
Open cylinder (no top): \(S = \pi r^2+2\pi r h\). So \(h = \dfrac{S-\pi r^2}{2\pi r}\). Volume: \(V = \pi r^2 h = \dfrac{r(S-\pi r^2)}{2}\).

Step 2: Differentiate and set to zero.
\(\dfrac{dV}{dr} = \dfrac{S-3\pi r^2}{2} = 0 \Rightarrow S = 3\pi r^2\).

Step 3: Find h in terms of r.
\(h = \dfrac{3\pi r^2-\pi r^2}{2\pi r} = \dfrac{2\pi r^2}{2\pi r} = r\). So \(h = r\), i.e. radius equals height.
\[ \boxed{r = h \text{ (radius equals height)}} \]