A metallic solid sphere of radius 9 cm is melted to form a solid cylinder of radius 9 cm. Find the height of the cylinder.

Answer

To solve the problem of finding the height of a cylinder formed by melting a metallic solid sphere, we can follow these steps:

### Step 1: Understand the relationship between the volumes
When a solid sphere is melted to form a solid cylinder, the volume of the sphere is equal to the volume of the cylinder.

### Step 2: Write the formula for the volume of the sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 3: Write the formula for the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the cylinder and \( h \) is the height of the cylinder.

### Step 4: Set the volumes equal to each other
Since the volume of the sphere is equal to the volume of the cylinder, we can write:
\[ \frac{4}{3} \pi R^3 = \pi r^2 h \]
where \( R \) is the radius of the sphere and \( r \) is the radius of the cylinder.

### Step 5: Substitute the known values
Given that the radius of the sphere \( R = 9 \) cm and the radius of the cylinder \( r = 9 \) cm, we substitute these values into the equation:
\[ \frac{4}{3} \pi (9)^3 = \pi (9)^2 h \]

### Step 6: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} (9)^3 = (9)^2 h \]

### Step 7: Calculate \( 9^3 \) and \( 9^2 \)
Calculating the powers:
\[ 9^3 = 729 \quad \text{and} \quad 9^2 = 81 \]
So the equation becomes:
\[ \frac{4}{3} \times 729 = 81h \]

### Step 8: Multiply and solve for \( h \)
Calculating \( \frac{4 \times 729}{3} \):
\[ \frac{2916}{3} = 972 \]
So we have:
\[ 972 = 81h \]

### Step 9: Divide both sides by 81 to find \( h \)
\[ h = \frac{972}{81} = 12 \text{ cm} \]

### Final Answer
The height of the cylinder is \( 12 \) cm.

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