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18. What is the locus of the points in a plane which are at a given distance from a given point not in the plane? What is the locus in space of all points: 19. Equidistant from two given points?

20. Equidistant from three given points?

21. Equidistant from two given parallel lines?

22. Equidistant from three given parallels not in the same plane?

23. Equidistant from two given intersecting lines?

24. Equidistant from the three edges of a trihedral angle?

25. At a given distance from a given plane?

26. Equidistant from two given parallel planes ? 27. Equidistant from two given intersecting planes? 28. Equidistant from the three faces of a trihedral angle? 29. What is the locus of all lines drawn through a given point, and parallel to a given plane?

30. What is the locus of points in a given plane which are equidistant from two given points not in the plane?

31. What is the locus of points equidistant from two given planes, and also equidistant from two given points?

32. What position relative to a given plane has a line if its projection on the plane is equal to its own length? What position if its projection is a point?

33. A line 4 feet long makes, with a plane, the angle 45°; find its projection on the plane.

34. If the projection of a line upon a plane is equal to half the length of the line, what is the angle of inclination of the line to the plane?

§ 2. THE PRISM.

1. One edge of a cube = a; find the surface, the volume, and the length of a diagonal.

2. Find one edge and the volume of a cube, if the sur96 sq. ft. Also find the same, if the surface

face

=

=

S.

3. Find the edge and the surface of a cube, if the volume = 1000 cub. ft. Also if the volume

=

=V.

4. How many square meters of surface require to be cemented, in constructing a cubical reservoir which will hold 8000kg of water?

5. Find the dimensions of a cubical block of marble which weighs 2700s. Specific gravity of marble = 2.7. 6. Find the volume of a cube, if a diagonal in one of its faces = a.

7. How much lead is used in lining the bottom and sides of a cubical vessel that holds 729 cub. ft. of water?

8. If a cubical vessel requires 320 sq. ft. of lead for lining the bottom and sides, how many cubic feet of water will it hold?

9. A plumber is ordered to make a cubical vessel which will hold 2 tons of water. What must be the length of one edge?

10. A cellar 12 ft. long and 9 ft. wide is flooded to a depth of 4 in. Find the weight of the water.

11. What is the weight of the air in a room 5m long, 4TM wide, and 3.2m high, if one liter of air weighs 1.293o?

12. How much lead will be required to line a cistern, open at the top, which is 4 ft. 6 in. long, 2 ft. 8 in. wide, and contains 42 cub. ft.?

13. How many bricks are required to build a wall 90 ft. long, 8 ft. high, and 18 in. thick, if the bricks are 9 in. long, 4 in. wide, and 3 in. thick?

14. A book is 8 in. long, 6 in. wide, and 14 in. thick. Find the depth of the box whose length and breadth are 3 ft. 4 in. and 2 ft. 6 in., that it may contain 400 such books.

15. An open cistern is made of iron 2 in. thick. The inner dimensions are: length, 4 ft. 6 in.; breadth, 3 ft.; depth, 2 ft. 6 in. What will the cistern weigh (i.) when empty? (ii.) when full of water? Specific gravity of iron = 7.2.

16. An open cistern 6 ft. long and 4 ft. wide holds 108 cub. ft. of water. How many cubic feet of lead will it take to line the sides and bottom, if the lead is in. thick?

17. Rain has fallen to the depth of half an inch; how many cubic feet of water have fallen on an acre?

18. The three dimensions of a rectangular parallelopiped are a, b, c; find the surface, the volume, and the length of a diagonal.

19. The volume of a parallelopiped = V, and the three dimensions are as m:n:p; find the dimensions.

Find the lateral surface and the volume of the following regular prisms, if in each case the height = h, and one side of the base =α:

20. Triangular prism.

21. Quadrangular prism.

22. Hexagonal prism.

23. Octagonal prism.

24. Find the volume of a right triangular prism, if its height is 14 in., and the sides of the base are 6, 5, and 5 in.

25. Find the volume of a right prism 14 ft. high, whose bases are squares, each side measuring 1 ft. 6 in.

26. The base of a right prism is a rhombus, one side of which 10 in., and the shorter diagonal = 12 in. The height of the prism 15 in. Find the entire surface and

=

the volume.

=

27. Find the volume of a right hexagonal prism whose height is 10 ft., each side of the hexagon being 10 in.

28. Find the volume of a right prism 32 ft. long, if its ends are trapezoids, the parallel sides of which are 12 ft. and 8 ft., and the perpendicular distance between them is 6 ft.

29. Find the volume of a right prism, if the height=3", and the base is a square the diagonal of which = 2.

30. How many cubic inches of mahogany will be required to veneer the top of a table in the shape of a regular hexagon, each side of which measures 2 ft., being of an inch thick?

the veneer

31. How many cubic feet of stone are required to build a dam 1000 ft. long, 20 ft. high, 10 ft. wide at the bottom, and 4 ft. wide at the top?

32. The wall of China is 1500 miles long, 20 ft. high, 15 ft. wide at the top, and 25 ft. wide at the bottom; how many cubic yards of material does it contain?

33. The distance around a reservoir in the shape of a regular hexagon is 360 ft. If the average monthly loss from evaporation amounts to a layer of water 2 in. deep, how many cubic feet of water must be supplied monthly to maintain the water at a constant level?

34. Given the volume V and the height h of a regular hexagonal prism, find one side a of the base.

§3. THE CYLINDER OF REVOLUTION.

1. How many square feet of sheet iron are required to make a funnel 2 ft. in diameter and 40 ft. long?

2. A right cylinder is 10 ft. high, and measures 7 ft. 4 in. around the base. Find the convex surface and the volume.

3. Find the radius of the base of a right cylinder if the volume = 1540 cu. in., and the height: = 10 in.

4. Find the height of a right cylinder if the volume= 3080 cub. ft., and the radius of the base = 7 ft.

5. I wish to have made a cylindrical pail 14 in. high, and holding exactly 4 cub. ft.; what must be the radius of the base?

6. If the total surface of a right cylinder closed at both ends is a, and the radius of the base is r, what is the height of the cylinder?

7. If the lateral surface of a right cylinder is a, and the volume is b, find the radius of the base and the height.

8. By how much is the volume of a right cylinder increased if the radius of the base is doubled? if the height is doubled? if both are doubled?

9. What will it cost to dig a well 3 ft. in diameter and 30 ft. deep, at $4.00 per cubic yard of earth thrown out?

10. How many cubic yards of earth must be removed in constructing a tunnel 100 yds. long, whose section is a semicircle with a radius of 10 ft.?

11. If the diameter of a well is 7 ft., and the water is 10 ft. deep, how many gallons of water are there, reckoning 7 gals. to the cubic foot?

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