Geometry Area, Volume and Dissection Arguments Test Solutions and Scoring (PDF)

\(2\pi r = 18\pi\), so \(r = 9\).

\(A = \pi \cdot 81 = 81\pi\)

r = 9; area 81π

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

\(V = \pi \cdot 5^{2} \cdot 12 = 300\pi = 942.5\)

942.5 cubic units

Scoring: 1 point for the formula, 1 point for the answer.

\(V = \frac{1}{3}\pi \cdot 9^{2} \cdot 12 = 324\pi\)

324π

Scoring: 1 point for the formula, 1 point for the answer.

Radius 9.

\(V = \frac{4}{3}\pi \cdot 9^{3} = 972\pi\)

972π

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

\(V = \frac{1}{3} \cdot 144 \cdot 10 = 480\)

480 cubic units

Scoring: 1 point for the base area, 1 point for the volume.

Cylinder: \(\pi \cdot 4^{2} \cdot 7 = 112\pi\).

Cavalieri’s principle: the other solid has the same volume, \(112\pi\).

112π each

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

\(\frac{1}{3}\pi \cdot 36 \cdot h = 48\pi\)

\(12h = 48\), so \(h = 4\).

h = 4

Scoring: 1 point for the equation, 1 point for h.

1. \(\frac{1}{3} \cdot 16 \cdot 4 = \frac{64}{3}\)
2. Three, since \(64 \div \frac{64}{3} = 3\).
3. Three equal pyramids fill the prism with the same base and height, so each is one third of Bh.

Scoring: 1 point per part.

Cylinder: \(\pi \cdot 25 \cdot 8 = 200\pi\).

Hemisphere: \(\frac{2}{3}\pi \cdot 125 = \frac{250}{3}\pi\).

\(200\pi + \frac{250}{3}\pi = \frac{850}{3}\pi = 890.1\)

890.1 cubic units

Scoring: 1 point for each solid, 1 point for the total.