Find the volume of the sector of a sphere of radius 25 cm. The radius of the base of the conical base is 7 cm.

Options

A. `344sqrt2 pi`
B. `(1280)/3 pi`
C. `120sqrt3 pi`
D. `(1250)/3 pi`

Answer

To find the volume of the sector of a sphere with a radius of 25 cm and a conical base radius of 7 cm, we will follow these steps:

### Step 1: Identify the given values
- Radius of the sphere (R) = 25 cm
- Radius of the base of the cone (r) = 7 cm

### Step 2: Use the Pythagorean theorem to find the height of the sector
We can visualize the problem as a right triangle where:
- The hypotenuse is the radius of the sphere (R = 25 cm).
- One leg is the radius of the cone (r = 7 cm).
- The other leg is the height (h) of the sector.

Using the Pythagorean theorem:
\[ R^2 = r^2 + h^2 \]
Substituting the values:
\[ 25^2 = 7^2 + h^2 \]
\[ 625 = 49 + h^2 \]
\[ h^2 = 625 - 49 \]
\[ h^2 = 576 \]
\[ h = \sqrt{576} = 24 \text{ cm} \]

### Step 3: Calculate the height of the sector
Since the height of the sector is the difference between the radius of the sphere and the height we just calculated:
\[ \text{Height of the sector} = R - h = 25 - 24 = 1 \text{ cm} \]

### Step 4: Use the formula for the volume of the sector of a sphere
The formula for the volume of a sector of a sphere is:
\[ V = \frac{2}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the sphere (25 cm)
- \( h \) is the height of the sector (1 cm)

Substituting the values into the formula:
\[ V = \frac{2}{3} \pi (25)^2 (1) \]
\[ V = \frac{2}{3} \pi (625) \]
\[ V = \frac{1250}{3} \pi \]

### Step 5: Final volume calculation
The volume of the sector of the sphere is:
\[ V \approx \frac{1250}{3} \pi \text{ cm}^3 \]

### Conclusion
Thus, the volume of the sector of the sphere is approximately \( \frac{1250}{3} \pi \) cm³.

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