Multiple choice

The volume of the largest cylinder that can be inscribed in a sphere of radius $'r'\ cm$ is (in cubic units)

  1. $\dfrac{4\pi r^3}{3\sqrt{3}}$
  2. $\dfrac{4\pi r^3 }{3\sqrt{2}}$
  3. $\dfrac{\pi r^3}{3\sqrt{2}}$
  4. $\dfrac{4\pi r^3}{2\sqrt{3}}$
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A Correct answer
Explanation

As derived in a previous question, the volume of the largest cylinder inscribed in a sphere of radius r is 4*pi*r^3 / (3*sqrt(3)).

AI explanation

Let h be the cylinder height and x be the base radius; using the right triangle relation inside the sphere, x squared plus one quarter of h squared equals r squared. Maximizing the cylinder volume yields a height of 2 times r divided by the square root of 3. Substituting this back into the volume formula gives 4 pi r cubed divided by 3 times the square root of 3 cubic units.

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