A solid metallic sphere of radius 43 cm is melted and recast into a right circular cylinder of radius 6 cm. What is the height (in cm) of the cylinder (correct to one decimal place)?

Options

A. (A) 1.8
B. (B) 2.1
C. (C) 3
D. (D)2.7

Answer

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

### Step 1: Calculate the volume of the sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given the radius of the sphere is 43 cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (43)^3 \]

### Step 2: Calculate \( 43^3 \)
First, we need to calculate \( 43^3 \):
\[ 43^3 = 43 \times 43 \times 43 = 79507 \]

### Step 3: Substitute \( 43^3 \) into the volume formula
Now substitute \( 79507 \) into the volume formula:
\[ V = \frac{4}{3} \pi (79507) \]

### Step 4: Calculate the volume of the sphere
Using \( \pi \approx 3.14 \):
\[ V \approx \frac{4}{3} \times 3.14 \times 79507 \]
Calculating this gives:
\[ V \approx \frac{4 \times 3.14 \times 79507}{3} \approx \frac{998,887.76}{3} \approx 332,962.59 \text{ cm}^3 \]

### Step 5: Set the volume of the cylinder equal to the volume of the sphere
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the cylinder and \( h \) is the height.

Given the radius of the cylinder is 6 cm, we can set up the equation:
\[ 332,962.59 = \pi (6^2) h \]

### Step 6: Calculate \( 6^2 \)
Calculating \( 6^2 \):
\[ 6^2 = 36 \]

### Step 7: Substitute \( 6^2 \) into the volume formula for the cylinder
Substituting this into the equation gives:
\[ 332,962.59 = \pi (36) h \]

### Step 8: Solve for \( h \)
Rearranging the equation to solve for \( h \):
\[ h = \frac{332,962.59}{\pi \times 36} \]
Using \( \pi \approx 3.14 \):
\[ h \approx \frac{332,962.59}{3.14 \times 36} \approx \frac{332,962.59}{113.04} \approx 2945.99 \text{ cm} \]

### Step 9: Convert to the correct units
Since we need the height in cm, we round to one decimal place:
\[ h \approx 2946.0 \text{ cm} \]

### Final Answer
The height of the cylinder is approximately **2946.0 cm**.