Question

name
pd
geometry module 1 end (2d and 3d figures)

1. a cylinder with a diameter of 7 inches and a height of 8 inches is shown. which two - dimensional figure below can be rotated about the axis to generate the cylinder? a. figure with dimensions 7 inches and 3.5 inches b. figure with dimensions 8 inches and 7 inches c. figure with dimensions 4 inches and 3.5 inches d. figure with dimensions 8 inches and 3.5 inches
2. a right triangle and a line on the coordinate plane are shown. select the objects generated by rotations of the triangle about x = 4 and about the x - axis. multiple figures with checkboxes
3. which object cannot be generated by the rotation of a circle or a semi - circle about a vertical line? a. sphere figure b. torus figure c. cylinder figure d. hemisphere figure
4. a right rectangular pyramid is shown below. what is the exact volume, in cubic units? hand - written answer 8.772 cubic units

Explanation:

1.

Step1: Recall cylinder - generation principle

A cylinder is generated by rotating a rectangle about an axis. The height of the cylinder is the length of the side of the rectangle parallel to the axis of rotation, and the radius of the base of the cylinder is the length of the side of the rectangle perpendicular to the axis of rotation. Given a cylinder with diameter \(d = 7\) inches (radius \(r=3.5\) inches) and height \(h = 8\) inches, we need a rectangle with one - side equal to the height of the cylinder and the other side equal to the radius of the base of the cylinder.

Step2: Analyze options

Option A has sides \(7\) inches and \(3.5\) inches which is incorrect. Option B has sides \(8\) inches and \(7\) inches which is incorrect. Option C has sides \(4\) inches and \(3.5\) inches which is incorrect. Option D has a rectangle with height \(8\) inches and width \(3.5\) inches. When this rectangle is rotated about an axis, it will generate the given cylinder.

Step1: Understand rotation of circles and semi - circles

A sphere is generated by rotating a semi - circle about its diameter (a vertical line in this case). A torus (a doughnut - shaped object) can be generated by rotating a circle about a vertical line that is outside the circle. A hemisphere is generated by rotating a semi - circle about its diameter. A cylinder cannot be generated by rotating a circle or a semi - circle about a vertical line. A cylinder is generated by rotating a rectangle about an axis.

Step1: Recall volume formula for a rectangular pyramid

The volume formula for a rectangular pyramid is \(V=\frac{1}{3}lwh\), where \(l\) is the length of the base, \(w\) is the width of the base, and \(h\) is the height of the pyramid. Here, \(l = 2.4\), \(w=2.15\), and \(h = 1.7\).

Step2: Calculate the volume

\[

$$\begin{align*} V&=\frac{1}{3}\times2.4\times2.15\times1.7\\ &=\frac{1}{3}\times(2.4\times2.15\times1.7)\\ &=\frac{1}{3}\times(5.16\times1.7)\\ &=\frac{1}{3}\times8.772\\ & = 2.924 \end{align*}$$

\]

Answer:

D. A rectangle with height 8 inches and width 3.5 inches

2.

• Rotation about \(x = 4\):
• When the right - triangle is rotated about the vertical line \(x = 4\), it will generate a cone with a circular base and a vertex. The object generated is a cone, so option A is correct.
• Rotation about the \(x\) - axis:
• When the right - triangle is rotated about the \(x\) - axis, it will generate a cone. The object generated is a cone, so option C is correct.