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1. How many square feet in the convex surface of a right prism whose altitude is 2 feet, and whose base is a regular hexagon of which each side is 8 inches long? How many square feet in the whole surface?
2. The radius of the base of a cylinder is 6 inches, and its altitude 3 feet; how many square feet in the whole surface?
3. What is the number of feet in the bounding planes of a cube whose edge is 5 feet? The number of solid feet in the cube?
4. What is the number of feet in the bounding planes of a right parallelopiped whose three dimensions are 4, 7, and 9 feet? The number of cubic feet in the parallelopiped?
5. What is the number of cubic feet in the right prism whose dimensions are given in the first example?
6. What is the number of cubic feet in the cylinder whose dimensions are given in the second example?
7. The altitude of a prism is 9 feet and the perimeter of the base 6 feet. What is the altitude and perimeter of the base of a similar prism one third as great?
8. What is the ratio of the volumes of two cylinders whose altitudes are as 3: 6?
9. How many square feet in the convex surface of a right pyramid whose slant height is 3 feet, and whose base is a regular octagon of which each side is 2 feet long?
10. How many square feet in the convex surface of a cone whose slant height is 5 feet and whose base has a radius of 2 feet? How many square feet in the whole surface?
11. How many cubic feet in a right quadrangular pyramid whose altitude is 10 feet, and whose base is 3 feet square?
12. How many cubic feet in the cone whose dimensions are given in the tenth example?
13. The slant height of a frustum of a right pyramid is 6 feet, and the perimeters of the two bases are 18 feet and 12 feet respectively; what is the convex surface of the frustum ?
14. What would be the slant height of the pyramid whose frustum is given in the preceding example?
15. What is the whole surface of a frustum of a cone whose altitude is 8 feet, and of whose bases the radii are 11 feet and 5 feet respectively?
16. The altitude of a pyramid is 25 feet, and its base is a rectangle 8 feet by 6; how many cubic feet in the pyramid ?
17. The altitude of a cone is 20 feet, and the radius of its base 5 feet; how many cubic feet in the cone?
18. How many cubic feet in a frustum of the cone given in the preceding example, cut off by a plane 5 feet from the base?
19. How far from the base must a cone whose altitude is 12 feet be cut off so that the frustum shall be equivalent to one half of the cone?
20. How many square feet in the surface of a sphere whose radius is 6 feet?
21. How many cubic feet in a sphere whose radius is 8 feet?
22. What is the ratio of the volumes of two spheres whose radii are as 4: 8?
23. Are spheres always similar solids? Are cones?
24. What is the least number of planes that can enclose a space?
66. The convex surfaces of prisms or pyramids of equal altitudes are as the perimeters of their bases. (14.)
67. The opposite faces of a parallelopiped are equal and parallel.
68. The four diagonals of a parallelopiped bisect each other.
69. A plane passing through the opposite edges of a parallelopiped bisects the parallelopiped.
70. In a right parallelopiped the diagonals are equal; and the square of each is equal to the sum of the squares of the three dimensions.
71. In a cube the square of a diagonal is three times the square of an edge.
72. Prisms are to each other as the products of their bases by their altitudes. (25.)
73. Prisms with equivalent bases are as their altitudes; with equal altitudes, as their bases. (72.)
74. Polygons formed by parallel planes cutting a pyramid are as the squares of their distances from the vertex. (39; II. 31.)
75. Pyramids are to each other as the products of their bases by their altitudes. (51.)
76. Pyramids with equivalent bases are as their altitudes; with equal altitudes, as their bases. (75.)
77. How can Theorem VIII. be proved from Theorem IX.?
78. If a pyramid is cut by a plane parallel to its base, the pyramid cut off will be similar to the whole pyramid. (39; 4).
79. In a sphere great circles bisect each other.
80. A great circle bisects a sphere. (54.)
81. The centre of a small circle is in the perpendicular from the centre of the sphere to the small circle.
82. Small circles equally distant from the centre of a sphere are equal.
83. The intersection of the surfaces of two spheres is the circumference of a circle.
81. The arc of a great circle can be made to pass through any two points on the surface of a sphere. (IV. 4.)
85. Definition. A plane is tangent to a sphere when it touches but does not cut the sphere.
86. Prove that the radius of a sphere to the point of tangency of a plane is perpendicular to the plane. (IV. 8.)
87. As the semi-decagon revolves about A F, what kind of a solid is described by the triangle ABK? What by the trapezoid K C? By LD?
88. The surface described by the line A B AKX circ. GO.
Draw from G a perpendicular to A B, and from the point where it meets A B a perpendicular to A F. (42.)
89. The surface described by the line CD L M × circ. GO.
90. Definition. The surfaces described by the arcs A B, BC, CD, &c. are called zones.
91. The area of a zone is equal to the product of its altitude by the circumference of a great circle.
92. Zones on the same or equal spheres are as their altitudes.
93. The surface of a sphere is four times the surface of one of its great circles. (62; III. 32.)
94. Definition. A polyedron is circumscribed about a sphere when its faces are each tangents to the sphere. In this case the sphere is inscribed in the polyedron.
95. The surface of a sphere is equal to the convex surface of the circumscribed cylinder. (62; 15.)
96. Definition. A Spherical Sector is the solid described by any sector of a semicircle as the semicircle revolves about its diameter.
97. The volume of a spherical sector is equal to the product of the surface of the zone forming its base by one third of the radius of the sphere of which it is a part.
98. A Spherical Segment is a part of a sphere included by two parallel planes cutting or touching the sphere. When one plane touches and one cuts the sphere, the spherical segment is called a spherical segment of one base; when both cut, a spherical segment of two bases.
99. How can the volume of a spherical segment of one base be found? A spherical segment of two bases?
100. A sphere is two thirds of the circumscribed cylinder.
101. A cone, hemisphere, and cylinder having equal bases and the same altitude are as the numbers 1, 2, 3.
PROBLEMS OF CONSTRUCTION.
IN the preceding demonstrations we have assumed that our figures were already constructed. The Problems of Construction given in this Book depend for their solution upon the principles of the preceding Books. In some of the problems the construction and demonstration are given in full; in others the construction is given and the propositions necessary to prove the construction referred to in the order in which they are to be used, and the pupil must complete the demonstration. In a few instances references are made to the Exercises appended to the previous Books. In such cases either the propositions to which reference is made can be demonstrated or the problem omitted.
1. To bisect a given straight line.
Let A B be the given straight line. From A and B as centres with a radius greater
than half of A B, describe arcs cutting one another at C and D; join C and D cutting AB at E, and the line AB is bisected at E. For C and D being each equally distant from A and B, the line CD must be perpendicular to AB at its middle point (converse of I. 53).