A hemisphere of iron is melted and recast in the shape of a right circular cylinder of diameter 18 cm and height 162 cm. The radius of the hemisphere is

Options

A. 27 cm
B. 9 cm
C. 6 cm
D. 12 cm

Answer

To find the radius of the hemisphere that is melted and recast into a right circular cylinder, we can follow these steps:

### Step 1: Understand the relationship between the volumes
When the hemisphere is melted and recast into a cylinder, the volume of the hemisphere will be equal to the volume of the cylinder.

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

### 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: Find the radius of the cylinder
Given the diameter of the cylinder is 18 cm, we can find the radius \( R \):
\[ R = \frac{\text{Diameter}}{2} = \frac{18}{2} = 9 \text{ cm} \]

### Step 5: Substitute the values into the volume equations
Now we can set the volumes equal to each other:
\[ \frac{2}{3} \pi r^3 = \pi (9^2)(162) \]

### Step 6: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ \frac{2}{3} r^3 = 9^2 \times 162 \]
Calculating \( 9^2 \):
\[ 9^2 = 81 \]
Now substituting back:
\[ \frac{2}{3} r^3 = 81 \times 162 \]

### Step 7: Calculate \( 81 \times 162 \)
Calculating \( 81 \times 162 \):
\[ 81 \times 162 = 13122 \]
So we have:
\[ \frac{2}{3} r^3 = 13122 \]

### Step 8: Solve for \( r^3 \)
To isolate \( r^3 \), multiply both sides by \( \frac{3}{2} \):
\[ r^3 = 13122 \times \frac{3}{2} = 19683 \]

### Step 9: Find the cube root of \( r^3 \)
Now, take the cube root of both sides to find \( r \):
\[ r = \sqrt[3]{19683} \]
Calculating the cube root:
\[ r = 27 \text{ cm} \]

### Final Answer
The radius of the hemisphere is \( 27 \) cm.

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