A parallelepiped is formed by planes drawn parallel to coordinate axes through the points A=(1,2,3) and B=(9,8,5). The volume of that parallelepiped is equal to (in cubic units)

Options

A. 192
B. 48
C. 32
D. 96

Explanation

To find the volume of the parallelepiped formed by the points A and B, we can follow these steps:

### Step 1: Identify the coordinates of points A and B
Given:
- Point A = (1, 2, 3)
- Point B = (9, 8, 5)

### Step 2: Translate the coordinates
To simplify the calculation, we can translate the coordinates so that point A becomes the origin (0, 0, 0). We do this by subtracting the coordinates of A from the coordinates of B.

New coordinates for point B:
- B' = B - A = (9 - 1, 8 - 2, 5 - 3)
- B' = (8, 6, 2)

### Step 3: Calculate the volume of the parallelepiped
The volume \( V \) of a parallelepiped formed by the vectors corresponding to the coordinates of point B' is given by the formula:
\[ V = |x \cdot y \cdot z| \]
where \( x, y, z \) are the components of the vector B'.

Substituting the values:
- \( x = 8 \)
- \( y = 6 \)
- \( z = 2 \)

Now, calculate the volume:
\[ V = 8 \times 6 \times 2 \]
\[ V = 48 \times 2 \]
\[ V = 96 \]

### Final Answer
The volume of the parallelepiped is \( 96 \) cubic units.

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