| Geometry Demystified, 2nd edition |
| Stan Gibilisco |
| Explanations for Quiz Answers in Chapter 8 |
| 1. All cubes have six faces. The correct choice is B. |
| 2. According to the formula on page 211 of the text, the volume of an ellipsoid varies in direct proportion to the product of the lengths of the three semiaxes. If we double the lengths of two semiaxes while leaving the third one unchanged, we increase the product of all three lengths by a factor of 2 x 2, or 4. The correct choice is B. |
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3. Let's review the formula for calculating the lateral surface area of a right circular cone, based on the radius r and the height h. We find that formula on page 202 of the text. If we call the lateral surface area S2, then S2 = π r (r2 + h2)1/2 In Fig. 8-14, r = 2 and h = 3, and we can consider both values mathematically exact. If we set π = 3.14159, then S2 = 3.14159 x 2 x (22
+ 32)1/2 rounded off to the nearest hundredth of a square unit. (We ran out the intermediate values for π and the square root of 13 to a few extra digits to avoid so-called cumulative rounding errors in the calculation process. That's always a good idea when doing long arithmetic calculations with approximate values.) The correct choice is C. |
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4. On page 203 of the text, we find a formula for calculating the volume of a right circular cone. If V represents the volume, r represents the radius, and h represents the height, then V = π r2 h / 3 In Fig. 8-14, r = 2 (exactly) and h = 3 (exactly). We can substitute these values into the foregoing formula to get V = π x 22 x 3 / 3 The correct choice is C. |
| 5. We can calculate the volume of a parallelepiped by multiplying the base area times the height, according to the formula on pages 199-200. In the scenario of Fig. 8-15, we can't figure out the base area because we don't know the angle between the sides measuring 7 and 4 units. Therefore, the correct response is choice D, "We need more information to answer this question." |
| 6. Once again, the volume of a parallelepiped equals the base area times the height. Whatever the base area happens to be, we can have complete confidence that if we triple the height, we'll triple the volume as well. The answer is A. |
| 7. A cube has six faces, all of which are identical squares. If we quadruple the surface area of a cube, then we "automatically" quadruple the interior area of each square face. In order to quadruple the interior area of a square, we must double the lengths of all four sides. Therefore, we quadruple the surface area of a cube if and only if we double the lengths of all the edges. Doubling the length of each edge of a cube (that is, increasing that length by a factor of 2) results in a volume increase of 23, or 8, for the whole cube. The correct choice is C. |
| 8. The surface area of a sphere varies in direct proportion to the square of its radius. If we quadruple a sphere's surface area (increase it by a factor of 4), then we "automatically" double its radius (increase it by a factor of 41/2, or 2). The volume of a sphere varies in direct proportion to the cube of its radius. If we double a sphere's radius by quadrupling its surface area, therefore, we increase the volume by a factor of 23, or 8. The correct choice is C. |
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9. Whenever we see a right circular cylinder or a slant circular cylinder, we can find its volume by taking π times the square of the radius, and then multiplying by the height. In the scenario of Fig. 8-16, the radius equals 6 units (exactly), while the height equals 10 units (exactly). If we call the volume V and we let π = 3.14159, then V =
3.14159 x 62 x 10 rounded off to the nearest cubic unit. The answer is C. |
| 10. The volume of a circular cylinder varies in direct proportion to the square of the radius (of the base and the top), assuming that the height stays constant. If we triple the cylinder's radius (increase it by a factor of 3), we increase the volume by a factor of 32, or 9. The correct choice is B. |