Cross Sections, Solids of Rotation and Geometric Modeling: Geometry Math Test Answer Key – PDF Download

Question 1: Name the cross sections (3 points)

(a) A circle. (b) A triangle (isosceles). (c) A circle.

Answers: circle; triangle; circle.

Scoring: 1 point for each correct shape.

Question 2: Cut through a cylinder (2 points)

(a) A rectangle \(12\text{ cm}\) wide (the diameter) and \(15\text{ cm}\) tall.

(b) Area \(=12\times 15=180\text{ cm}^{2}\).

Answers: rectangle \(12\text{ cm}\) by \(15\text{ cm}\); \(180\text{ cm}^{2}\).

Scoring: 1 point for the shape with both dimensions; 1 point for the area.

Question 3: Slicing a sphere (2 points)

(a) \(x^{2}+6^{2}=10^{2}\), so \(x^{2}=64\) and \(x=8\text{ cm}\).

(b) Area \(=\pi(8)^{2}=64\pi\text{ cm}^{2}\).

Answers: \(8\text{ cm}\); \(64\pi\text{ cm}^{2}\).

Scoring: 1 point for the radius (right triangle with hypotenuse 10); 1 point for the area.

Question 4: Rotating a triangle (3 points)

(a) A cone.

(b) The leg on the axis is the height, \(12\text{ cm}\), and the other leg is the radius, \(5\text{ cm}\).

(c) \(V=\frac{1}{3}\pi(5)^{2}(12)=100\pi\text{ cm}^{3}\).

Answers: cone; \(r=5\text{ cm}\), \(h=12\text{ cm}\); \(100\pi\text{ cm}^{3}\).

Scoring: 1 point for each part.

Question 5: Comparing counties (2 points)

(a) A: \(31{,}200: 40=780\). B: \(24{,}640: 28=880\) people per square mile.

(b) County B is denser by \(880-780=100\) people per square mile.

Answers: 780 and 880; County B by 100.

Scoring: 1 point for both densities; 1 point for the comparison with the difference.

Question 6: Heating a gym (2 points)

(a) Volume \(=50\times 30\times 12=18{,}000\text{ ft}^{3}\). BTUs \(=18{,}000\times 4=72{,}000\).

(b) \(72{,}000: 18{,}000=4\) hours.

Answers: 72,000 BTUs; 4 hours.

Scoring: 1 point for each part.

Question 7: Cheaper box (3 points)

(a) \(10\times 6\times 6=360\) and \(9\times 8\times 5=360\).

(b) Box 1: \(2(60+60+36)=312\text{ in}^{2}\). Box 2: \(2(72+45+40)=314\text{ in}^{2}\).

(c) Box 1: \(312\times 0.03=\$9.36\). Box 2: \(314\times 0.03=\$9.42\). Box 1 costs less.

Answers: both \(360\text{ in}^{3}\); \(312\text{ in}^{2}\) and \(314\text{ in}^{2}\); Box 1 at \$9.36.

Scoring: 1 point for each part.

Question 8: Grain silo (3 points)

(a) \(V=\pi(5)^{2}(12)=300\pi\text{ m}^{3}\).

(b) \(V=\frac{1}{2}\cdot\frac{4}{3}\pi(5)^{3}=\frac{250\pi}{3}\text{ m}^{3}\).

(c) Total \(=300\pi+\frac{250\pi}{3}=\frac{1150\pi}{3}\approx 1{,}204.3\text{ m}^{3}\).

Answers: \(300\pi\); \(\frac{250\pi}{3}\); about \(1{,}204.3\text{ m}^{3}\).

Scoring: 1 point for each part.

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