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A solid toy is in the form of a hemisphere surmounted by a right circular cone. Ratio of the radius of the cone to its slant height is $3: 5$. If the volume of the toy is $240\pi$ cm$^3$, then find the total height of the toy.
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Let the radius and the slant height of the cone be $3x$ cm and $5x$ cm respectively
$\therefore$ height of the cone (h) $= \sqrt{(5x)^2 - (3x)^2} = 4x$ cm
According to question, volume of toy $= 240\pi$
$\frac{2}{3}\pi(3x)^3 + \frac{1}{3}\pi(3x)^2(4x) = 240\pi$
Solving, we get $x = 2$
$\therefore$ Total height of toy $= [4(2) + 3(2)]$ cm $= 14$ cm
$\therefore$ height of the cone (h) $= \sqrt{(5x)^2 - (3x)^2} = 4x$ cm
According to question, volume of toy $= 240\pi$
$\frac{2}{3}\pi(3x)^3 + \frac{1}{3}\pi(3x)^2(4x) = 240\pi$
Solving, we get $x = 2$
$\therefore$ Total height of toy $= [4(2) + 3(2)]$ cm $= 14$ cm