Question

The plane region is revolved completely about the​ x-axis to sweep out a solid of revolution. Describe the solid AND find its volume in terms of π.

Answer 1

Answer:

C

Step-by-step explanation:

what happens, when you rotate something around a pole ? what shape is the most outside point making around that pole ?

a circle, of course.

that means rotating the rectangle around the x-axis creates a round object (with the bottom area being a circle).

so, all answers about a prism (rectangular bottom shape) are automatically wrong.

again, just imagine to rotate a small sheet of paper around a pole (the x-axis acts like one in this scenario).

the result is a solid with a circular bottom area, and it maintains this shape all the way to the top.

it is a cylinder !

and what is its radius ?

well, the outermost point of the rectangle is 8 points away from the x-axis. when we rotate this rectangle, that attribute remains, of course. and every point on the side surface of the resulting cylinder has the same distance from the central x-axis : 8.

so, the radius is 8.

and the volume of a cylinder is

ground area × height = pi×r² × height

in our case

pi×8² × 5 (the width or "height" of the rectangle) = pi×64 × 5 = 320×pi