Final answer:
To find the volume of the solid obtained by rotating the triangle about the x-axis, we can use the theorem of Pappus. The volume is equal to the area of the triangle times the distance traveled by its centroid. Using the given vertices, the formula for the area of a triangle, and the formulas for the coordinates of the centroid, we can calculate the volume.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the triangle about the x-axis, we can use the theorem of Pappus. The theorem states that the volume of a solid of revolution is equal to the product of the area of the generating region and the length of the path traveled by its centroid. In this case, we need to find the area of the triangle and the distance traveled by its centroid.
To find the area of the triangle, we can use the formula for the area of a triangle as 0.5 times the absolute value of the cross product of two of its sides. We can use the vertices (1, 2), (1, 6), and (4, 4) to determine the lengths of the sides. Using the formula, we get:
Area = 0.5 * |(1 - 1)(4 - 1) - (2 - 6)(4 - 1)| = 4 square units
The centroid of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be found using the following formulas:
x-coordinate of centroid = (x1 + x2 + x3) / 3
y-coordinate of centroid = (y1 + y2 + y3) / 3
Using the vertices (1, 2), (1, 6), and (4, 4), we can calculate the coordinates of the centroid:
x-coordinate of centroid = (1 + 1 + 4) / 3 = 2
y-coordinate of centroid = (2 + 6 + 4) / 3 = 4
The distance traveled by the centroid when rotating the triangle about the x-axis is equal to the y-coordinate of the centroid, which is 4 units.
Now, we can apply the theorem of Pappus to find the volume:
Volume = Area * Distance traveled by centroid
Volume = 4 square units * 4 units = 16 cubic units
Therefore, the volume of the solid obtained by rotating the triangle about the x-axis is 16 cubic units.