The material of a sphere of radius r is melted and recast into a hollow cylindrical shell of thickness a and outer radius b. What is its length assuming that no material is lost in recasting?

Options

A. `(4 r^(3))/(3a(2a-b))`
B. `(4r^(3))/(3a(b+a)(b-a))`
C. `(4r^(3))/(3a(b-a))`
D. `(4r^(3))/(3a(2b-a))`

Answer

To solve the problem of finding the length of a hollow cylindrical shell formed from a melted sphere, we can follow these steps:

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

### Step 2: Define the Dimensions of the Hollow Cylinder
The hollow cylindrical shell has:
- Outer radius \( b \)
- Thickness \( a \)

From this, we can determine the inner radius \( r_{\text{inner}} \) of the cylinder:
\[ r_{\text{inner}} = b - a \]

### Step 3: Calculate the Volume of the Hollow Cylinder
The volume \( V_{\text{cylinder}} \) of a hollow cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi (b^2 - (b - a)^2) \cdot L \]
Expanding the inner term:
\[ (b - a)^2 = b^2 - 2ab + a^2 \]
Thus,
\[ V_{\text{cylinder}} = \pi (b^2 - (b^2 - 2ab + a^2)) \cdot L = \pi (2ab - a^2) \cdot L \]

### Step 4: Set the Volumes Equal
Since no material is lost during the recasting, we set the volume of the sphere equal to the volume of the hollow cylinder:
\[ \frac{4}{3} \pi r^3 = \pi (2ab - a^2) L \]

### Step 5: Cancel Out \(\pi\) and Rearrange for Length \( L \)
Cancelling \(\pi\) from both sides gives:
\[ \frac{4}{3} r^3 = (2ab - a^2) L \]
Now, solving for \( L \):
\[ L = \frac{\frac{4}{3} r^3}{2ab - a^2} \]

### Final Expression for Length
Thus, the length \( L \) of the hollow cylindrical shell is:
\[ L = \frac{4r^3}{3(2ab - a^2)} \]