Multiple choice

The radius of cylinder of maximum volume which can be inscribed in a right circular cone of radius $R$ and height $H$ (axis of cylinder and cone are same) is given by

  1. $\dfrac{R}{2}$
  2. $\dfrac{R}{3}$
  3. $\dfrac{2R}{3}$
  4. $\dfrac{2R}{5}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a cylinder of radius r and height h inscribed in a cone of radius R and height H, r/R = (H-h)/H. Volume V = pi * r^2 * h = pi * r^2 * H * (1 - r/R). Setting dV/dr = 0 gives r = 2R/3.

AI explanation

Let the inscribed cylinder have radius r and height h. By comparing the similar triangles formed by the cross-section of the cone and cylinder, the relation is h = H - (H/R) * r. The volume of the cylinder is V = pi * r^2 * h, so substituting for h gives V = pi * r^2 * (H - (H/R) * r). Differentiating this volume with respect to r and setting it to zero yields H * (2r - 3r^2 / R) = 0, which gives the maximizing radius as r = 2R/3.

Keep practicing — related questions