Question

1. a 3 - d solid is formed by rotating the rectangle shown at the right about the vertical line. the dimensions of the rectangle are stated in inches. what is the volume of the solid formed by the rotation, in cubic inches? a) 90 in³ b) 45 in³ c) 90π in³ d) 45π in³ 10. which figure can have the same cross - section as a sphere? a) description of first 3 - d figure b) description of second 3 - d figure c) description of third 3 - d figure d) description of fourth 3 - d figure

Explanation:

Step1: Identify the solid formed by rotation

When a rectangle with length $l = 5$ inches and width $w=3$ inches is rotated about a vertical - line, a cylinder is formed. The radius of the cylinder $r$ is equal to the width of the rectangle ($r = 3$ inches) and the height of the cylinder $h$ is equal to the length of the rectangle ($h = 5$ inches).

Step2: Use the volume formula for a cylinder

The volume formula for a cylinder is $V=\pi r^{2}h$. Substitute $r = 3$ and $h = 5$ into the formula: $V=\pi\times3^{2}\times5=\pi\times9\times5 = 45\pi$ cubic inches.

For question 10:
The cross - section of a sphere is a circle. When a cone is cut parallel to its base, the cross - section is a circle. A rectangular prism (options a and b) and a triangular prism (option c) do not have circular cross - sections in general.

Answer:

1. d) $45\pi$ in$^{3}$
2. d)