Final answer:
Rotating the triangle about line segment AB creates a cone. The height (h) is 2 units, and the radius (r) is 5 units. Thus, the volume of the cone is approximately 157.1 cubic units.
Step-by-step explanation:
When you rotate a triangle 360° about one of its sides, you generate a cone. In this case, the line segment AB serves as the axis of rotation. Since vertices A and B lie on this axis, they are fixed points during the rotation, and vertex C describes a circle, creating a conical shape with the circle as its base. To calculate the volume of this cone, you need the radius of the base and the height of the cone.
For the given triangle with vertices A(-4, 2), B(-2, -2), and C(-2, 3), AB is the height of the cone, and the perpendicular distance from C to AB is the radius of the cone's base. The distance between B and C can be found using the distance formula, giving the height (h) of the cone as:
h = √((-2 - (-4))² + (-2 - 2)²) = √(4), which simplifies to 2 units.
The radius (r) of the cone's base is the distance from C to AB. Since AB and the x-axis are parallel, the y-values of B and C provide the vertical distance (which is the radius). Therefore, r = 3 - (-2) = 5 units.
Now, the volume (V) of the cone can be found using the formula V = (1/3)πr²h. By substituting the known values of r and h, we have V = (1/3)π(5)²(2) ≈ 157.1 cubic units. After rounding to the nearest tenth, the volume is 157.1 cubic units.