The height of the cylinder of maximum volume inscribed in a sphere of radius '$a$' is
- $\dfrac { 3a }{ 2 } $
- $\dfrac { \sqrt { 2 } a }{ 3 } $
- $\dfrac { a }{ \sqrt { 3 } } $
- $\dfrac{2a}{\sqrt{3}} $
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Correct answer
Explanation
Volume of cylinder V = pi * r^2 * h. In a sphere of radius a, r^2 + (h/2)^2 = a^2, so r^2 = a^2 - h^2/4. V = pi * (a^2 - h^2/4) * h = pi * (a^2*h - h^3/4). Differentiating with respect to h: dV/dh = pi * (a^2 - 3h^2/4) = 0. 3h^2/4 = a^2, h^2 = 4a^2/3, h = 2a/sqrt(3).
AI explanation
For a cylinder of maximum volume inscribed in a sphere of radius a, let the cylinder's height be h and its radius be r. The sphere's radius relates to the cylinder's dimensions by a^2 = r^2 + (h/2)^2. Maximizing the cylinder's volume V = pi*r^2*h by substituting r^2 = a^2 - h^2/4 yields a maximum when h^2 = 4a^2/3. Therefore, the height is 2a/sqrt(3).