Question

a triangle with coordinates (2, 0), (6, 0), and (2, 2) is rotated 360° around the x - axis. what is the resulting three - dimensional figure?
a cone with radius 2 and height 4
a cone with radius 4 and height 2
a triangular prism with side length 2 and height 4
a triangular prism with side length 4 and height 2

Explanation:

Step1: Analyze the triangle coordinates

The triangle has vertices \((2,0)\), \((6,0)\), and \((2,2)\). When rotated around the \(x -\)axis, we can use the disk/washer method conceptually (but for basic shape - recognition). The base of the triangle along the \(x -\)axis from \(x = 2\) to \(x=6\) (length \(b=4\)) and height \(h = 2\) (in the \(y -\)direction).

Step2: Recall the formula for the volume of a cone

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\). When a right - triangle (which is the case here, since two sides are parallel to the axes) is rotated around one of its legs (the \(x -\)axis, where the leg is along the \(x -\)axis from \((2,0)\) to \((6,0)\)), the resulting solid is a cone. The radius \(r\) of the cone is the length of the non - \(x\) - axis leg. Here, \(r = 2\) (the \(y -\)coordinate of the point \((2,2)\)), and the height \(h\) of the cone is the length of the \(x\) - axis leg, \(h=4\).

Answer:

a cone with radius \(2\) and height \(4\)