A sphere of 5 cm radius is cut by two parallel planes which are 7 cm apart but on the opposite sides of the centre of the sphere. The radius of one of the ends of the zone is 3 cm.
What is the volume (in cu. cm.) of the zone?

Options

A. `(434)/3 pi`
B. `(112)/3 pi`
C. `(541)/3 pi`
D. `121 pi`

Answer

To find the volume of the zone formed by the two parallel planes cutting through the 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. Here, the radius \( r = 5 \) cm.

\[ V = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \text{ cm}^3 \]

### Step 2: Determine the height of the spherical caps
The distance between the two parallel planes is 7 cm. Since these planes are on opposite sides of the center of the sphere, the height of each spherical cap can be calculated as follows:

1. The radius of the sphere is 5 cm.
2. The distance from the center of the sphere to one of the planes is \( 5 - 3 = 2 \) cm (since the radius of one end of the zone is 3 cm).
3. Therefore, the height of the spherical cap above the plane with radius 3 cm is \( 5 - 2 = 3 \) cm.
4. The height of the spherical cap below the plane is \( 5 - 4 = 1 \) cm (since the total distance between the planes is 7 cm, and the height of the cap above is 3 cm, the height below must be \( 7 - 3 = 4 \) cm).

### Step 3: Calculate the volume of the spherical caps
The volume \( V \) of a spherical cap is given by the formula:

\[ V = \frac{1}{3} \pi h^2 (3R - h) \]

where \( h \) is the height of the cap and \( R \) is the radius of the sphere.

#### Volume of the upper cap (height = 4 cm):
\[ V_1 = \frac{1}{3} \pi (4)^2 (3 \times 5 - 4) = \frac{1}{3} \pi (16)(15 - 4) = \frac{1}{3} \pi (16)(11) = \frac{176}{3} \pi \text{ cm}^3 \]

#### Volume of the lower cap (height = 2 cm):
\[ V_2 = \frac{1}{3} \pi (2)^2 (3 \times 5 - 2) = \frac{1}{3} \pi (4)(15 - 2) = \frac{1}{3} \pi (4)(13) = \frac{52}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the volume of the zone
The volume of the zone is the volume of the sphere minus the volumes of the two caps:

\[ V_{zone} = V_{sphere} - (V_1 + V_2) \]

Substituting the values:

\[ V_{zone} = \frac{500}{3} \pi - \left(\frac{176}{3} \pi + \frac{52}{3} \pi\right) \]

\[ V_{zone} = \frac{500}{3} \pi - \frac{228}{3} \pi = \frac{272}{3} \pi \text{ cm}^3 \]

### Final Answer
The volume of the zone is:

\[ \frac{272}{3} \pi \text{ cm}^3 \]