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 , then find the total height of the toy.

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Solution

Q.
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 $240 π$ ${cm}^{3}$ , then find the total height of the toy.

Ans.
(14)

Let the radius and the slant height of the cone be $3 x$ cm and $5 x$ cm respectively

$∴$ height of the cone $(h) = \sqrt{{(5 x)}^{2} - {(3 x)}^{2}} = \sqrt{25 x^{2} - 9 x^{2}} = \sqrt{16 x^{2}} = 4 x cm$

According to question, volume of toy = $240 π$

$\Rightarrow \frac{2}{3} π {(3 x)}^{3} + \frac{1}{3} π {(3 x)}^{2} (4 x) = 240 π$

$\Rightarrow \frac{2}{3} π \times 27 x^{3} + \frac{1}{3} π \times 36 x^{3} = 240 π$

$\Rightarrow \frac{1}{3} π \times 9 x^{3} (6 + 4) = 240 π$

$\Rightarrow \frac{1}{3} π \times 9 x^{3} \times 10 = 240 π$

$\Rightarrow x^{3} = 8 \Rightarrow x = 2$

$∴ Total height of toy = [4 (2) + 3 (2)] cm = 14 cm$

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