A solid metalic hemisphere of radius 8 cm is melted and recast into a right circular cone of base radius 6 cm. Determine the height of the cone.

Answer

To solve the problem of determining the height of the cone formed by melting a solid metallic hemisphere, we will follow these steps:

### Step 1: Calculate the volume of the hemisphere.
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere.

Given that the radius \( r = 8 \) cm, we can substitute this value into the formula:
\[ V = \frac{2}{3} \pi (8)^3 \]
Calculating \( (8)^3 \):
\[ (8)^3 = 512 \]
Now substituting this back into the volume formula:
\[ V = \frac{2}{3} \pi (512) = \frac{1024}{3} \pi \, \text{cm}^3 \]

### Step 2: Set up the equation for the volume of the cone.
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

Given that the base radius \( r = 6 \) cm, we can substitute this value into the formula:
\[ V = \frac{1}{3} \pi (6)^2 h \]
Calculating \( (6)^2 \):
\[ (6)^2 = 36 \]
Thus, the volume of the cone becomes:
\[ V = \frac{1}{3} \pi (36) h = 12 \pi h \, \text{cm}^3 \]

### Step 3: Equate the volumes of the hemisphere and the cone.
Since the hemisphere is melted and recast into the cone, their volumes are equal:
\[ \frac{1024}{3} \pi = 12 \pi h \]

### Step 4: Solve for the height \( h \).
We can cancel \( \pi \) from both sides of the equation:
\[ \frac{1024}{3} = 12h \]
Now, multiply both sides by 3 to eliminate the fraction:
\[ 1024 = 36h \]
Now, divide both sides by 36 to solve for \( h \):
\[ h = \frac{1024}{36} \]
Simplifying \( \frac{1024}{36} \):
\[ h = \frac{256}{9} \approx 28.44 \, \text{cm} \]

### Final Answer:
The height of the cone is \( \frac{256}{9} \) cm or approximately \( 28.44 \) cm.

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