A solid sphere of radius 9 cm is melted to form a sphere of radius 6 cm and a right circular cylinder of same radius. The height of the cylinder so formed is ?
19 cm
To solve this problem, we need to use the concept of conservation of volume. The volume of the original solid sphere will be equal to the combined volume of the new sphere and the cylinder. Let's break it down:
Step 1: Calculate the volume of the original sphere.
Formula for the volume of a sphere: \( V = \frac{4}{3}\pi r^3 \).
Radius of the original sphere, \( r = 9 \) cm.
\( V_{\text{original}} = \frac{4}{3}\pi (9)^3 = \frac{4}{3}\pi \times 729 = 972\pi \text{ cm}^3 \).
Step 2: Calculate the volume of the new sphere.
Radius of the new sphere, \( r = 6 \) cm.
\( V_{\text{new sphere}} = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi \times 216 = 288\pi \text{ cm}^3 \).
Step 3: Determine the remaining volume for the cylinder.
The volume of the cylinder, \( V_{\text{cylinder}} = V_{\text{original}} - V_{\text{new sphere}} = 972\pi - 288\pi = 684\pi \text{ cm}^3 \).
Step 4: Calculate the height of the cylinder.
Formula for the volume of a cylinder: \( V = \pi r^2 h \).
Radius of the cylinder, \( r = 6 \) cm.
Using the volume, \( 684\pi = \pi (6)^2 h \).
Solving for height \( h \):
\( 684 = 36h \).
\( h = \frac{684}{36} = 19 \text{ cm} \).
Therefore, the height of the cylinder is 19 cm.
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