Question

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius \(= \sqrt{3}\) is:

• A. \(\frac{4}{3}\sqrt{3}\pi\)
• B. \(\frac{8}{3}\sqrt{3}\pi\)
• C. \(4\pi\)
• D. \(2\pi\)

Hint:

Relate radius r and height h of the cylinder by \(r^2 + h^2/4 = 3\), then maximise \(V = \pi r^2h\).

Answer 1

The Correct Option is C

Let the cylinder have radius r and height h. Then \(r^2 + \frac{h^2}{4} = 3\).
\(V = \pi r^2h = \pi\left(3 - \frac{h^2}{4}\right)h = \pi\left(3h - \frac{h^3}{4}\right)\).
\(\frac{dV}{dh} = \pi\left(3 - \frac{3h^2}{4}\right) = 0\) gives \(h = 2\); \(\frac{d^2V}{dh^2} = -\frac{3\pi h}{2} \lt 0\) (maximum).
Then \(r^2 = 3 - 1 = 2\) and \(V = \pi\cdot2\cdot2 = 4\pi\).
Answer: (3) \(4\pi\)