The position vector `oversetto(OA)` of point A `(7m,3m,0)` is rotated about `y-`axis. Find the volume `("in" m^(3))` of the cone thus formed `:("Take": pi=22/7)`.

Options

A. `154 m^(3)`
B. `66m^(3)`
C. `22m^(3)`
D. None

Explanation

To find the volume of the cone formed by rotating the position vector of point A (7m, 3m, 0) about the y-axis, we can follow these steps:

### Step 1: Identify the dimensions of the cone
When the point A (7m, 3m, 0) is rotated about the y-axis, it forms a cone. The height (h) of the cone is the y-coordinate of point A, and the radius (r) of the cone is the x-coordinate of point A.

- Height (h) = 3m (y-coordinate)
- Radius (r) = 7m (x-coordinate)

### Step 2: Write the formula for the volume of a cone
The formula for the volume (V) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

### Step 3: Substitute the values into the formula
Now, we will substitute the values of r and h into the volume formula. We also need to use the value of π as given in the question, which is \( \frac{22}{7} \).

\[ V = \frac{1}{3} \times \frac{22}{7} \times (7)^2 \times (3) \]

### Step 4: Calculate \( r^2 \)
Calculate \( r^2 \):

\[ r^2 = 7^2 = 49 \]

### Step 5: Substitute \( r^2 \) back into the volume formula
Now substitute \( r^2 \) back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \times 3 \]

### Step 6: Simplify the equation
Now simplify the equation:

1. The 3 in the numerator and denominator cancels out:
\[ V = \frac{22}{7} \times 49 \]

2. Calculate \( 22 \times 49 \):
\[ 22 \times 49 = 1078 \]

3. Now divide by 7:
\[ V = \frac{1078}{7} = 154 \]

### Final Result
Thus, the volume of the cone formed is:

\[ V = 154 \, \text{m}^3 \]