Geometry Cross Sections and Solids of Revolution Test Solutions and Scoring (PDF)

A rectangle 8 by 9.

\(8 \cdot 9 = 72\)

Rectangle; area 72

Scoring: 1 point for the shape, 1 point for the area.

The rectangle is 4 by \(4\sqrt{2}\).

\(4 \cdot 4\sqrt{2} = 16\sqrt{2} = 22.6\)

16√2, about 22.6

Scoring: 1 point for the dimensions, 1 point for the area.

1. A circle.
2. Base 12 and height 8: \(\frac{1}{2} \cdot 12 \cdot 8 = 48\).
3. Half of the radius: 3.

(a) Circle; (b) 48; (c) 3

Scoring: 1 point per part.

\(r = \sqrt{10^{2} – 6^{2}} = 8\).

\(A = 64\pi\)

Radius 8; area 64π

Scoring: 1 point for the radius, 1 point for the area.

1. Cylinder, \(V = \pi \cdot 25 \cdot 12 = 300\pi\)
2. Cone, \(V = \frac{1}{3}\pi \cdot 25 \cdot 12 = 100\pi\)
3. Sphere, \(V = \frac{4}{3}\pi \cdot 216 = 288\pi\)

Scoring: 1 point per part (0.5 for the solid and 0.5 for the volume).

The cross section is similar to the base with ratio \(\frac{4}{12} = \frac{1}{3}\).

Side \(\frac{10}{3}\), area \(\frac{100}{9}\).

Side 10/3; area 100/9

Scoring: 1 point for the ratio, 1 point for side and area.

Inner radius 2, outer radius 5, height 7.

\(\pi(5^{2} – 2^{2}) \cdot 7 = 147\pi\)

147π

Scoring: 1 point for the radii, 1 point for the volume.

(1) circle: a sphere cut anywhere. (2) triangle: a cone cut through its apex and axis. (3) rectangle: a cylinder cut through its axis.

Scoring: 2 points for all three, 1 point for two correct.

Each face is a circle. One piece has thickness \(24 \div 12 = 2\) cm.

\(V = \pi \cdot 3^{2} \cdot 2 = 18\pi = 56.5\)

Circle; 56.5 cubic centimeters

Scoring: 1 point for the shape and thickness, 1 point for the volume.