Question

The slant height of a right circular cone is $\sqrt{3}\text{ cm}$. The height of the cone for maximum volume is

• A. $\sqrt{5}\text{ cm}$
• B. $5\text{ cm}$
• C. $\sqrt{3}\text{ cm}$
• D. $1\text{ cm}$

Hint:

For any right circular cone with a fixed slant height $l$, the volume is always maximized at a unique fixed ratio: $h = \frac{l}{\sqrt{3}}$. Substituting our given value $l = \sqrt{3}$ into this ratio gives $h = \frac{\sqrt{3}}{\sqrt{3}} = 1\text{ cm}$ instantly!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A right circular cone has fixed slant height l = √3 cm; find the vertical height h that maximizes its volume.

Step 2: Key Formula or Approach:
Volume V = (1/3)πr²h. Using l² = r² + h² → r² = l² – h² = 3 – h². Express V in terms of h, set dV/dh = 0, check d²V/dh²<0.

Step 3: Detailed Explanation:
V = (π/3)(3h – h³). dV/dh = (π/3)(3 – 3h²) = π(1 – h²). Setting to zero: 1 – h² = 0 → h = 1 cm (positive). d²V/dh² = –2πh<0 at h=1, confirming maximum.

Step 4: Final Answer:
The height for maximum volume is 1 cm, matching option (D).