EXERCISES. GROUP 74 PROBLEMS CONCERNING THE SPHERE Ex. 1. At a given point on a sphere, construct a plane tangent to the sphere. Ex. 2. Through a given point on the surface of a sphere, draw an are of a great circle perpendicular to a given arc. Ex. 3. Inscribe a circle in a given spherical triangle. Ex. 4. Construct a spherical triangle, given its polar triangle. Given the radius, 7, construct a spherical surface which shall pass through Ex. 5. Three given points. Ex. 6. Two given points and be tangent to a given plane. Given the radius, r, construct a spherical surface which shall be tangent to Ex. 10. Three given planes. Ex. 11. Two given planes and one given sphere. Ex. 12. Construet a spherical surface which shall pass through three given points and be tangent to a given plane. Ex. 13. Through a given straight line pass a plane tangent to a given sphere. [SUG. Through the center of the sphere pass a plane 1 given line, etc.] When is the solution impossible? Ex. 14. Through a given point on a sphere, construct an arc of a great circle tangent to a given small circle of the sphere. [SUG. Draw a straight line from the center of the sphere to the given point, and produce it to intersect the plane of the small circle, etc.] NUMERICAL EXERCISES IN SOLID GEOMETRY For methods of facilitating numerical computations, see Arts. 493-6. EXERCISES. CROUP 75 LINES AND SURFACES OF POLYHEDRONS Find the lateral area and total area of a right prism whose Ex. 1. Base is an equilateral triangle of edge 4 in., and whose altitude is 15 in. Ex. 2. Base is a triangle of sides 17, 12, 25, and whose altitude is 20. Ex. 3. Base is an isosceles trapezoid, the parallel bases being 10 and 15 and leg 8, and whose altitude is 24. Ex. 4. Base is a rhombus whose diagonals are 12 and 16, and whose altitude is 12. Ex. 5. Base is a regular hexagon with side 8 ft., and whose altitude is 20 ft. Ex. 6. Find the entire surface of a rectangular parallelopiped 8 X 12 X 16 in.; of one pX & Xr ft. Ex. 7. Of a cube whose edge is 1 ft. 3 in. Ex. 8. The lateral area of a regular hexagonal prism is 120 sq. ft. and an edge of the base is 10 ft. Find the altitude. Ex. 9. How many square feet of tin are necessary to line a box 20 X 6X4 in. ? Ex. 10. If the surface of a cube is 1 sq. yd., find an edge in inches Ex. 11. Find the diagonal of a cube whose edge is 5 in. Ex. 12. If the diagonal of a cube is 12 ft., find the surface. (469) Ex. 13. If the surface of a rectangular parallelopiped is 208 sq. in., and the edges are as 2:3:4, find the edges. In a regular square pyramid Ex. 14. If an edge of the base is 16 and slant height is 17, find the altitude. Ex. 15. If the altitude is 15 and a lateral edge is 17, find an edge of the base. Ex. 16. If a lateral edge is 25 and an edge of the base is 14, find the altitude. In a regular triangular pyramid Ex. 17. If an edge of the base is 8 and the altitude is 10, find the slant height. Ex. 18. Find the altitude of a regular tetrahedron whose edge is 6. Find the lateral surface and the total surface of Ex. 19. A regular square pyramid an edge of whose base is 16, and whose altitude is 15. Ex. 20. A regular triangular pyramid an edge of whose base is 10, and whose altitude is 12. Ex. 21. A regular hexagonal pyramid an edge of whose base is 4, and whose altitude is 21. Ex. 22. A regular square pyramid whose slant height is 24, and whose lateral edge is 25. Ex. 23. A regular tetrahedron whose edge is 4. Ex. 24. A regular tetrahedron whose altitude is 9. Ex. 25. A regular hexagonal pyramid each edge of whose base is a, and whose altitude is b. Ex. 26. In the frustum of a regular square pyramid the edges of the bases are 6 and 18, and the altitude is 8. Find the slant height. Hence find the lateral area. Ex. 27. In the frustum of a regular triangular pyramid the edges of the bases are 4 and 6, and the altitud. is 5. Find the slant height. Hence find the lateral area. Ex. 28. In the frustum of a regular tetrahedron, if the edge of the lower base is b1, the edge of the upper base is b2, and the altitude is a, show that L= √ } ( b1 — b2 )2 +4a2. 2 Ex. 29. In the frustum of a regular square pyramid the edges of the bases are 20 and 60, and a lateral edge is 101. Find the lateral surface. EXERCISES. GROUP 78 LINES AND SURFACES OF CONES AND CYLINDERS Ex. 1. How many square feet of lateral surface has a tunnel 100 yds. long and 7 ft. in diameter Ex. 2. The lateral area of a cylinder of revolution is 1 sq. yd., and the altitude is 1 ft. Find the radius of the base. Ex. 3. The entire surface of a cylinder of revolution is 900 sq. ft. and the radius of the base is 10 ft. Find the altitude. In a cylinder of revolution Ex. 4. Find R in terms of S and H. Ex. 5. Find H in terms of R and T. Ex. 6. Find Tin terms of S and H. Ex. 7. How many sq. yds, of canvas are required to make a conical tent 20 ft. in diameter and 12 ft. high? Ex. 8. A man has 400 sq. yds. of canvas and wants to make a conical tent 20 yds. in diameter. What will be its altitude? Ex. 9. The altitude of a cone of revolution is 10 ft. and the lateral area is 11 times the area of the base. Find the radius of the base. In a cone of revolution Ex. 10. Find Tin terms of S and L. Ex. 11. Find R in terms of T and L. Ex. 12. How many square feet of tin are necessary to make a funnel the diameters of whose ends are 2 in. and 8 in., and whose altitude is 7 in. ? Ex. 13. If the slant height of a frustum of a cone of revolution makes an angle of 45° with the base, show that the lateral area of the frustum is (r2—r22) π √2. 2 EXERCISES. GROUP 77 SPHERICAL LINES AND SURFACES Ex. 1. Find in square feet the area of the surface of a sphere whose radius is 1 ft. 2 in. Ex. 2. How many square inches of leather will it take to cover a baseball whose diameter is 34 in. ? Ex. 3. How many sq. ft. of tin are required to cover a dome in the shape of a hemisphere 6 yds. in diameter ? Ex. 4. What is the radius of a sphere whose surface is 616 sq. in. ? Ex. 5. Find the diameter of a globe whose surface is 1 sq. yd. Ex. 6. If the circumference of a great circle on a sphere is 1 ft., find the area of the surface of the sphere. Ex. 7. If a hemispherical dome is to contain 100 sq. yds. of surface, what must its diameter be? Ex. 8. Find the radius of a sphere in which the area of the surface equals the number of linear units in the circumference of a great circle. Find the area of a lune in which Ex. 9. The angle of the lune is 36°, and the radius of the sphere is 14 in. Ex. 10. The angle of the lune is 18° 20′, and the diameter of the sphere is 20 in. Ex. 11. The angle of the lune is 24°, and the surface of the sphere is 4 sq. ft. Find the area of a spherical triangle in which Ex. 12. The angles are 80°, 90°, 120°, and the diameter of the sphere is 14 ft. Ex. 13. The angles are 74° 24′, 83° 16′, 92° 20′, and the radius of the sphere is 10. Ex. 14. The angles are 85°, 95°, 135°, and the surface of the sphere is 10 sq. ft. |