Question

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?

• A. 144
• B. 96
• C. 72
• D. 24

Answer 1

The Correct Option is C

To find out how many identical cones are formed when a solid hemisphere is melted, we need to determine the volumes of the hemisphere and a cone, and then divide the volume of the hemisphere by the volume of one cone.

Step 1: Calculate the volume of the solid hemisphere.

The formula for the volume of a hemisphere is:

\( V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \)

Given the radius of the hemisphere (\( r = 24 \, \text{cm} \)), the volume is:

\( V_{\text{hemisphere}} = \frac{2}{3} \pi (24)^3 = \frac{2}{3} \pi \times 13824 = 9216\pi \, \text{cm}^3 \)

Step 2: Calculate the volume of one cone.

The formula for the volume of a cone is:

\( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \)

Given the base radius of the cone (\( r = 8 \, \text{cm} \)) and the height (\( h = 6 \, \text{cm} \)), the volume is:

\( V_{\text{cone}} = \frac{1}{3} \pi (8)^2 (6) = \frac{1}{3} \pi \times 64 \times 6 = 128\pi \, \text{cm}^3 \)

Step 3: Determine the number of cones formed.

To find the number of cones, divide the volume of the hemisphere by the volume of one cone:

\( \frac{V_{\text{hemisphere}}}{V_{\text{cone}}} = \frac{9216\pi}{128\pi} = \frac{9216}{128} = 72 \)

Therefore, 72 identical cones are formed from the solid hemisphere.