Semivertical angle of a cone of maximum volume of given slant height is
- $\tan ^{ -1 }{ 2 } $
- $\cot ^{ -1 }{ 2 } $
- $\tan ^{ -1 }{ \sqrt { 2 } } $
- $\cot ^{ -1 }{ \sqrt { 2 } } $
Reveal answer
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C
Correct answer
Explanation
For a cone of fixed slant height l, the volume V = (1/3) * pi * r^2 * h. With r = l * sin(theta) and h = l * cos(theta), V = (1/3) * pi * l^3 * sin^2(theta) * cos(theta). Maximizing this leads to tan(theta) = sqrt(2).
AI explanation
Let the semivertical angle be theta and the constant slant height be l, so the cone radius is r = l sin theta and height is h = l cos theta. The volume V = (1/3)pi r^2 h = (1/3)pi l^3 sin^2 theta cos theta. Differentiating V with respect to theta and setting it to zero gives 2 sin theta cos^2 theta - sin^3 theta = 0, meaning tan^2 theta = 2. Therefore, the semivertical angle for maximum volume is tan inverse of square root of 2.