Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 5.
The Correct Answer and Explanation is :
The correct answer is: {\frac{125}{6}} \text{ cubic units}
To find the volume of the solid bounded by the planes (x = 0), (y = 0), (z = 0), and (x + y + z = 5), we can visualize this solid as a tetrahedron in the first octant of three-dimensional space. The equation (x + y + z = 5) represents a plane that intersects the axes at the points where (x), (y), or (z) equals 5.
Steps to Determine the Volume:
- Identify the Vertices:
- The plane (x + y + z = 5) intersects the axes at:
- ( (5, 0, 0) ) (when (y = 0) and (z = 0))
- ( (0, 5, 0) ) (when (x = 0) and (z = 0))
- ( (0, 0, 5) ) (when (x = 0) and (y = 0))
- Thus, the vertices of the tetrahedron are at the points ( (0, 0, 0) ), ( (5, 0, 0) ), ( (0, 5, 0) ), and ( (0, 0, 5) ).
- Volume Formula:
The volume (V) of a tetrahedron with vertices at the origin and points ( (a, 0, 0) ), ( (0, b, 0) ), ( (0, 0, c) ) is given by the formula:
[
V = \frac{1}{6} \times a \times b \times c
] - Plugging in the Values:
Here, (a = 5), (b = 5), and (c = 5):
[
V = \frac{1}{6} \times 5 \times 5 \times 5 = \frac{125}{6}
]
Conclusion:
The volume of the solid bounded by the given planes is:
[
\boxed{\frac{125}{6}} \text{ cubic units}
]
This result indicates that the solid occupies approximately (20.83) cubic units. Understanding this volume in relation to the first octant and how the tetrahedron is formed helps visualize the three-dimensional shape and the intersection of the planes. The tetrahedral shape is a fundamental geometric concept in calculus and can be extended to higher dimensions and more complex shapes.