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Seven metal hollow hemispheres of internal and external diameters 20 cm and 24 cm respectively are melted and recast into one solid cone of radius 28 cm. The height of the cone is ____ cm.

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Detailed Solution

Given, external diameters

=

24 cm, implies external radius (R) = 12 cm
Internal diameters = 20 cm, implies internal radius (r) = 10 cm
Also, radius of cone (

r^{'}) = 28 cm

Let h be the height of the cone
We know that,
Volume of a hollow hemisphere

= \frac{2}{3} π (R^{3} - r^{3})

Volume of cone

= \frac{1}{3} π ({r^{'}}^{2}) h

According to the question,

7 \times

\frac{2}{3} π (R^{3} - r^{3}) = \frac{1}{3} π r^{'} h

7 \times

\frac{2}{3} π (12^{3} - 10^{3}) = \frac{1}{3} π (28) (28) h

7 \times

2 (1728 - 1000) = (28) (28) h

7 \times

2 (728) = (28) (28) h

(728) = 2 (28) h

h = 13 cm
Therefore, the height of the cone is 13 cm.

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