Question

7 practice 7 (from unit 5, lesson 1)
if the two - dimensional figures are rotated around the vertical axes of rotation shown, what solids are formed?

Explanation:

Step1: Analyze the first figure (rectangle)

A rectangle rotated about a vertical axis (one of its sides) forms a cylinder. The formula for the volume of a cylinder is $V = \pi r^2 h$, but here we just need to identify the solid. The rectangle, when rotated around the vertical dashed line (which is one of its vertical sides), creates a three - dimensional shape with circular bases and a rectangular lateral surface, which is a cylinder.

Step2: Analyze the second figure (semicircle)

A semicircle, when rotated about its diameter (the vertical dashed line here), forms a sphere. The set of all points in space at a fixed distance (the radius of the semicircle) from a given point (the center of the semicircle) is a sphere. When we rotate the semicircle around its diameter, every point on the semicircle traces out a circle, and together they form a sphere.

Answer:

When the rectangle is rotated around the vertical axis, a cylinder is formed. When the semicircle is rotated around the vertical axis, a sphere is formed.