A solid spherical copper ball , whose diameter is 14 cm , is melted and converted into a wire having diameter equal to 14 cm. The length of the wire is

Options

A. 27 cm
B. `(16)/(3) cm`
C. 15 cm
D. `(28)/(3) cm`

Answer

To solve the problem of finding the length of the wire formed from a solid spherical copper ball, we will follow these steps:

### Step 1: Find the radius of the sphere
Given the diameter of the sphere is 14 cm, we can find the radius (r) using the formula:
\[ r = \frac{\text{diameter}}{2} \]
So,
\[ r = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \]

### Step 2: Calculate the volume of the sphere
The volume (V) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the radius we found:
\[ V = \frac{4}{3} \pi (7 \text{ cm})^3 \]
Calculating \( 7^3 \):
\[ 7^3 = 343 \]
So,
\[ V = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi \text{ cm}^3 \]

### Step 3: Find the radius of the wire
The diameter of the wire is also given as 14 cm, so the radius (R) of the wire is:
\[ R = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \]

### Step 4: Write the formula for the volume of the cylinder (wire)
The volume (V) of a cylinder is given by the formula:
\[ V = \pi R^2 h \]
Where \( h \) is the height (length) of the cylinder (wire). Substituting the radius:
\[ V = \pi (7 \text{ cm})^2 h \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
So,
\[ V = 49\pi h \text{ cm}^3 \]

### Step 5: Set the volumes equal to each other
Since the volume of the melted sphere is equal to the volume of the wire, we can set the equations equal:
\[ \frac{1372}{3} \pi = 49\pi h \]

### Step 6: Solve for h
We can cancel \( \pi \) from both sides:
\[ \frac{1372}{3} = 49h \]
Now, solve for \( h \):
\[ h = \frac{1372}{3 \times 49} \]
Calculating \( 3 \times 49 = 147 \):
\[ h = \frac{1372}{147} \]
Now, dividing:
\[ h \approx 9.33 \text{ cm} \]

### Conclusion
The length of the wire is approximately 9.33 cm.

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