A solid hemisphere of radius 24 cm is melted, and identical cones, each of base radius 8 cm and height 6 cm, are formed. How many such cones are formed?

Question

Accepted Answer

#### Step 1: Volume of the Hemisphere
The volume of a hemisphere is given by the formula:  
$$
V_{	ext{hemisphere}} = \frac{2}{3} \pi r^3
$$
Substitute $$ r = 24 \, 	ext{cm} $$:  
$$
V_{	ext{hemisphere}} = \frac{2}{3} \pi (24)^3 = \frac{2}{3} \pi (13824) = 9216 \pi \, 	ext{cm}^3
$$

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#### Step 2: Volume of One Cone
The volume of a cone is given by the formula:  
$$
V_{	ext{cone}} = \frac{1}{3} \pi r^2 h
$$
Substitute $$ r = 8 \, 	ext{cm} $$ and $$ h = 6 \, 	ext{cm} $$:  
$$
V_{	ext{cone}} = \frac{1}{3} \pi (8)^2 (6) = \frac{1}{3} \pi (64)(6) = 128 \pi \, 	ext{cm}^3
$$

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#### Step 3: Number of Cones Formed
The number of cones formed is the ratio of the volume of the hemisphere to the volume of one cone:  
$$
	ext{Number of cones} = \frac{V_{	ext{hemisphere}}}{V_{	ext{cone}}}
$$
Substitute the values:  
$$
	ext{Number of cones} = \frac{9216 \pi}{128 \pi} = 72
$$

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Thus, **72 cones** can be formed, making Option **(c)** correct.

Answer

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