Question 62

A solid is hemispherical at the bottom and conical above. If the surface areas of the two parts are equal, then the ratio of radius and height of its conical part is

Solution

let the radius of base = r

height of cone = h

$$ 2\pi r^2 = \pi r \sqrt r^2 + \sqrt h^2 $$

$$ 2r = \sqrt r^2 +\sqrt h^2 $$

$$ 4r^2 = r^2 + h^2 $$

$$ 3r^2 = h^2 $$

$$ h = \sqrt 3 r $$

$$ \frac{r}{h} = \frac{1}{\sqrt 3} $$

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