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The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is:
- a)60
- b)600
- c)720
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The volume of the tetrahedron included between the plane 3x + 4y - 5z ...
Given Information:
The equation of the plane is 3x + 4y - 5z - 60 = 0.
Volume of the Tetrahedron:
To find the volume of the tetrahedron included between the plane and the coordinate planes, we need to first find the coordinates of the points where the plane intersects the coordinate axes. These points will form the vertices of the tetrahedron.
Intercepts with Coordinate Axes:
To find the intercepts, we set y = z = 0 to find x-intercept, x = 20; x = y = 0 to find z-intercept, z = -12; x = z = 0 to find y-intercept, y = 15.
Volume Calculation:
The volume of the tetrahedron can be calculated using the formula V = (1/6) * Base Area * Height. Since the base of the tetrahedron is a triangle and the height is the distance between the plane and the origin, we can find the base area and height using the coordinates of the intercept points.
Base Area Calculation:
The base of the tetrahedron is a triangle with vertices (20, 0, 0), (0, 15, 0), and (0, 0, -12). We can calculate the area of this triangle using the formula for the area of a triangle given its vertices.
Height Calculation:
The height of the tetrahedron is the perpendicular distance from the origin to the plane. We can calculate this distance using the formula for the distance between a point and a plane.
Final Calculation:
Once we have the base area and height, we can plug these values into the volume formula V = (1/6) * Base Area * Height to find the volume of the tetrahedron.
Therefore, the correct answer is option B) 600.
The equation of the plane is 3x + 4y - 5z - 60 = 0.
Volume of the Tetrahedron:
To find the volume of the tetrahedron included between the plane and the coordinate planes, we need to first find the coordinates of the points where the plane intersects the coordinate axes. These points will form the vertices of the tetrahedron.
Intercepts with Coordinate Axes:
To find the intercepts, we set y = z = 0 to find x-intercept, x = 20; x = y = 0 to find z-intercept, z = -12; x = z = 0 to find y-intercept, y = 15.
Volume Calculation:
The volume of the tetrahedron can be calculated using the formula V = (1/6) * Base Area * Height. Since the base of the tetrahedron is a triangle and the height is the distance between the plane and the origin, we can find the base area and height using the coordinates of the intercept points.
Base Area Calculation:
The base of the tetrahedron is a triangle with vertices (20, 0, 0), (0, 15, 0), and (0, 0, -12). We can calculate the area of this triangle using the formula for the area of a triangle given its vertices.
Height Calculation:
The height of the tetrahedron is the perpendicular distance from the origin to the plane. We can calculate this distance using the formula for the distance between a point and a plane.
Final Calculation:
Once we have the base area and height, we can plug these values into the volume formula V = (1/6) * Base Area * Height to find the volume of the tetrahedron.
Therefore, the correct answer is option B) 600.
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Community Answer
The volume of the tetrahedron included between the plane 3x + 4y - 5z ...
The given equation of the plane is 3x + 4y − 5z − 60 = 0, it can be written in the form
which meets the coordinate axes at the points A(20, 0, 0), 8(0, 15, 0) and (0, 0, −12).
The coordinates of the origin are (0, 0, 0).
Therefore, volume of the tetrahedron OABC is
which meets the coordinate axes at the points A(20, 0, 0), 8(0, 15, 0) and (0, 0, −12).
The coordinates of the origin are (0, 0, 0).
Therefore, volume of the tetrahedron OABC is
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The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is:a)60b)600c)720d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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