Question

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

• A. Height of the cylinder
  % Telugu: స్థూపం ఎత్తుకు సమానం
• B. Height of the cylinder / 2
  % Telugu: స్థూపం ఎత్తు / 2
• C. 2 times Height of the cylinder
  % Telugu: స్థూపం ఎత్తుకు 2 రెట్లు
• D. 3 times Height of the cylinder % Telugu: స్థూపం ఎత్తుకు 3 రెట్లు

Hint:

Open cylinder (one end closed, one open): Surface Area \(S = \pi r^2 (\text{base}) + 2\pi rh (\text{curved surface})\). Volume \(V = \pi r^2 h\).
Express V as a function of one variable (e.g., r) using the constant S constraint.
Differentiate V with respect to that variable and set to zero to find critical points.
Use the second derivative test to confirm maximum.
For an open cylinder with given surface area and maximum volume, radius \(r\) = height \(h\).
(For a closed cylinder, \(r = h/2\) or \(h=2r\)).

Answer 1

The Correct Option is A

Problem: Maximize the volume of an open cylinder (closed at the base, open at the top) for a fixed surface area \( S \).

Let:

• \( r \): radius of the base
• \( h \): height of the cylinder

Surface Area (S) of the open cylinder:

\[ S = \pi r^2 + 2\pi r h \]

Solve for height \( h \):

\[ 2\pi r h = S - \pi r^2 \Rightarrow h = \frac{S - \pi r^2}{2\pi r} = \frac{S}{2\pi r} - \frac{r}{2} \]

Volume (V) of the cylinder:

\[ V = \pi r^2 h = \pi r^2 \left( \frac{S}{2\pi r} - \frac{r}{2} \right) = \frac{S r}{2} - \frac{\pi r^3}{2} \]

Differentiate V with respect to \( r \) to find the maximum:

\[ \frac{dV}{dr} = \frac{S}{2} - \frac{3\pi r^2}{2} \]

Set \( \frac{dV}{dr} = 0 \) for critical point:

\[ \frac{S}{2} - \frac{3\pi r^2}{2} = 0 \Rightarrow S = 3\pi r^2 \]

Substitute back to find \( h \):

\[ h = \frac{S}{2\pi r} - \frac{r}{2} = \frac{3\pi r^2}{2\pi r} - \frac{r}{2} = \frac{3r}{2} - \frac{r}{2} = r \]

So, at maximum volume:

\[ h = r \Rightarrow \text{Radius} = \text{Height} \]

Second derivative test:

\[ \frac{d^2V}{dr^2} = -3\pi r < 0 \quad \text{(for } r > 0 \text{)} \]

This confirms a maximum.

✅ Final Answer: \( \boxed{\text{Height of the cylinder}} \)