19. What is the surface of the largest cube that can be cut from a sphere which contains 14137.2 cu. ft.? Ans. 1800 sq. ft. 20. Two boys are flying a kite. The string is 720 feet long. One boy who stood directly under the kite, was 56 feet from the other boy who held the string; how high was the kite? Ans. 717.8+feet.

21. How many pounds of wheat in a cylindrical sack whose diameter is 11⁄2 feet, and whose length is 134 yards? (=3.1416) Ans. 447.31 lb.

22. How large a square can be cut from a circle 50 inches in diameter? Ans. 35.3553391 in.

23. How many bbl. in a tank in the form of the frustum of a pyramid, 5 feet deep, 10 feet square at the bottom and 9 feet square at the top? Ans. 107.26 bbl.

24. From a circular farm of 270 acres, a father gives to his sons equal circular farms, touching each other and the boundary of the farm. He takes for himself a circular portion in the center, equal in area to a son's part, and reserves the vacant tracts around his part for pasture lands and gives each son one of the equal spaces left along the boundary. Required the number of sons and the amount of pasture land each has.

Ans. 6 sons; 8.46079 A.

25. At each angle of a triangle being on a level plain and having sides respectively 40, 50, and 60 feet, stands a tower whose height equals the sum of the two sides including the angle. Required the length of a ladder to reach the top of each tower without moving at the base.

Ans. 116.680316+ft.

26. If the door of a room is 4 feet wide, and is opened to the angle of 90 degrees, through what distance has the outer edge of the door passed? Ans. 6.2832 feet..

27. A tinner makes two similar rectangular oil cans whose inside dimensions are as 3, 7, and 11. The first hold 8 gallons and the second being larger requires 4 times as much tin as the other. What are the dimensions of the smaller and the contents of the larger?

Dimensions of smaller 6, 14, and 22 inches.

Ans. Capacity of larger 64 gallons.

28. An 8-inch globe is covered with gilt at 8 cents per square inch; find Ans. $16.08.

the cost.

29. A hollow cylinder 6 feet long, whose inner diameter is 1 inch and outer diameter two inches, is transformed into a hollow sphere whose outer diameter is twice its inner diameter; find outer diameter. Ans. 3.59 in.

30. A circular field is 360 rods in circumference; what is the diagonal of a square field containing the same area? Ans. 20.3 rods.

31. What is the volume of a cylinder, whose length is 9 feet and the circumference of whose base is 6 feet? Ans. 25.78 cu. ft.

32. How many acres in a square field, the diagonal being 80 rods?

Ans. 20 acres.

33. How many cubical blocks, each edge of which is of a foot, will fill a box 8 feet long, 4 feet wide, and 2 feet thick. Ans. 1728 blocks.

34. From one corner of a rectangular pyramid 6 by 8 feet, it is 19 feet to the apex; find the dimentions of a rectangular solid whose dimensions are as 2, 3, and 4, that may be equivalent in volume. Ans. 4, 9, and 8 feet.

35.* A solid metal ball, 4 inches radius, weighs 8 lbs.; what is the thickness of spherical shell of the same metal weighing 75% lb., the external diameter of which is 10 inches? Ans. 1 inch.

36. What is the difference between 25 feet square and 25 square feet? Ans. 600 sq. ft.

37.* Find the greatest number of trees that can be planted on a lot 11 rods square, no two trees being nearer each other than one rod?

Ans. 152 trees.

38.* A straight line 200 feet long, drawn from one point in the outer edge of a circular race track to another point in the same, just touches the inner edge of the track. Find the area of the track and its width.

Ans. Area, Ta2=10000 sq. ft.; width, indeterminate.

39. The perimeter of a certain field in the form of an equilateral triangle is 360 rods; what is the area of the field? Ans. 543.552 sq. rd.

What length

40. A room is 18 feet long, 16 feet wide, and 10 feet high. of rope will reach from one upper corner to the opposite upper corner and touch the floor?

Ans. 35 3 ft.

41. How many bushels of wheat in a box whose length is twice its width, and whose width is 4 times its height; diagonal being 9 feet?

Ans. 25 bu., nearly. 42 Find the area of a circular ring whose breadth is 2 inches and inside diameter 9 inches. Ans. 69.1152 sq. in.

43* A round stick of timber 12 feet long, 8 inches in diameter at one end and 16 inches at the other, is rolled along till the larger end describes a complete circle. Required the circumference of the circle.

Ans. 150.83 feet.

44. A fly traveled by the shortest possible route from the lower corner to the opposite upper corner of a room 18 feet long, 12 feet wide and 10 feet high. Find the distance it traveled Ans. 28.42534 feet.

45. From the middle of one side and through the axis perpendicularly of a right triangular prism, sides 12 inches, I cut a hole 4 inches square. Find the volume removed. Ans. 138.564064 cu. in.

The base

46.* Two isosceles triangles have equal areas and perimeters. of one is 24 feet, and one of the equal sides of the other is 29 feet. The area of both is 10 times the area of a triangle whose sides are 13, 14, and 15 feet. Find the perimeters and altitudes.

Ans. Perimeters, 98 feet; altitudes 35 and 21 feet. 47. A grocer at one straight cut took off a segment of a cheese which had 14 of the circumference, and weighed 3 pounds; what did the whole weigh? Ans. 33.023 lb.

48.* A twelve inch ball is in a corner where walls and floor are at right angles; what must be the diameter of another ball which can touch that ball while both touch the same floor and the same walls?

Ans. 3.2154 in. or 44.7846 in.

49. What will it cost to paint a church steeple, the base of which is an octagon, 6 feet on each side, and whose slant height is 80 feet, at 30 cents per square yard? Ans. $64.

50. A tree 48 feet high breaks off; the top strikes the level ground 24 feet from the bottom of the tree; find the height of the stump. Ans. 18 feet.

51. How many acres in a square field whose diagonal is 54 rods longer than one of its sides? Ans. 160.6446 sq. rd.

52.* Three poles of equal length are erected on a plane so that their tops meet, while their bases are 90 feet apart, and distance from the point where the poles meet to the center of the triangle below is 65 feet. What is the length of the poles?

Ans. 83.23 feet.

53. A field contains 200 acres and is 5 times as long as wide. it cost to fence it, at a dollar per rod?

What will Ans. $960.

54.* What is the greatest number of plants that can be set on a circular piece of ground 100 feet in diameter, no two plants to be nearer each other than 2 feet and none nearer the circumference than 1 foot? Ans. 2173.

55. The axes of an ellipse are 100 inches and 60 inches; what is the difference in area between the ellipse and a circle having a diameter equal to the conjugate axis? Ans. 600 π-1884.96 sq. in.

56. Find the diameter of a circle of which the altitude of its greatest inscribed triangle is 25 feet. Ans. 33 feet.

57. If we cut from a cubical block enough to make each dimension 1 inch shorter, it will lose 1657 cubic inches, what are the dimensions?

58. Show that the area of a rhombus is one-half the rectangle formed by its diagonals. Noble Co. Ex. Test.

59. The length and breadth of a rectangular field are in the ratio of 4 to 3. How many acres in the field, if the diagonal is 100 rods?

60. A spherical vessel 30 inches in diameter contains in depth, 1 foot of water; how many gallons will it take to fill it? Holmes Co. Ex. Test. Ans. 39 gallons.

61. A field is 40 rods by 80 rods. How long a line from the middle of one end will cut off 71⁄2 acres? Ans. 80.6 rd., nearly. 62. A ladder 20 feet long leans against a perpendicular wall at an angle of 30°. How far is its middle point from the bottom of the wall?

Ans. 10 feet.

63. Four towers, A 125 feet high, B 75 feet, C 100 feet, and D 65 feet, stand on the same plane. B due south and 40 rods from A; C east of B and D south of C. The distance from A to C plus the distance from C to B is half a mile, and the distance from D to B is 82% yd. farther than the distance from C to D. What length of line is required to connect the tops of A and D? Ans. 240+rds.

64. Find the volume of the largest square pyramid that can be cut from a cone 9 feet in diameter and 20 feet high? Ans. 270 cu. ft.

65. A rectangular lawn 60 yd. long and 40 yd. wide has a walk 6 ft. wide around it and paths of the same width through it, joining the points of the opposite sides. Find in square yards the area of one of the four plats inclosed by paths. Ans. 459 sq. yd.

66. Which has the greater surface, a cube whose volume is 13.824 cu. ft., or a rectangular solid of equal volume whose length is twice its width, and its width twice its height? Ans. Rect. 576 sq. ft., more.

67. The volume of a rectangular tin can is 3 cu. ft. 1053 cu. in.; its dimensions are in the proportion of 11, 7, and 3. Find the area of tin in the Ans. 16% sq. ft.

can.

68. A conical well has a bottom diameter of 28 ft. 3 in., top diameter 56 ft. 6 in., and depth 23 ft. 1.2 in. Find its capacity in barrels.

Ans. 8023 bbl.

69. A cylindrical vessel 1 foot deep and 8 inches in diameter was 1 full of water; after a ball was dropped into the vessel it was full. Find the diameter of the ball. Ans. 6 inches.

70. Two logs whose diameters are 6 feet lie side by side. What is the diameter of a third log placed in the crevice on top of the two, if the pile is 9 feet high?. Ans. 4 ft.

71. Circles 6 and 10 feet in diameter touch each other; if perpendiculars from the center are let fall to the line tangent to both circles, how far apart will they be? Ans. 7.756 ft. 72. What are the linear dimensions of a rectangular box whose capacity is 65910 cubic feet; the length, breadth, and depth being to each other as 5, 3, and 2? Ans. 65, 39, and 26 ft.

73. The perimeter of a piece of land in the form of an equilateral triangle is 624 rods; what is the area? Ans. 117 A. 13 31 P.

74. Four logs 4 feet in diameter lay side by side and touch each other; on these and in the crevices lay three logs 3 feet in diameter; on these three and in the crevices lay two logs 2 feet in diameter; what is the diameter of a log that will lay on the top of the pile touching each of the logs 2 feet in diameter and the middle one of the logs 3 feet in diameter?

Ans.

75. What will it cost to gild a segment of a sphere whose diameter is 6 inches: the altitude of the segment being 2 inches, at 5 per square inch?

76. A grocer cut off the segment of a cheese, and circumference. What is the weight of the whole weighed 12 lbs?

found it took of the cheese, if the segment Ans. 52.0228+lbs.

They are in

77. Two ladders are standing in the street 20 feet apart. clined equally toward each other at the top, forming an angle of 45°. by arithmetic, the length of the ladders?

Find,

Ans. 26.13 ft. Union Co. Ex. List.

78. Two trees stand on opposite sides of a stream 120 feet wide; the height of one tree is to the width of the stream as 5 is to 4, and the width of the stream is to the height of the other as 5 is to 4; what is the distance between their tops? Ans. 131.58-ft.

79. How many gallons of water will fill a circular cistern 6 feet deep and 4 feet in diameter? Ans. 564.0162 gal.

80. A cube of silver, whose diagonal is 6 inches, was evenly plated with gold; if 4 cubic inches of gold were used, how thick was the plating? Ans. in.

81. Required the distance between the lower corner and the opposite upper corner of a room 60 feet long, 32 feet wide, and 51 feet high?

Ans 85 ft. 82. How deep must be a rectangular box whose base inside is 4 inches by 4 inches to hold a quart, dry measure?

Ans. 4.2 cu. in 30 feet long, 20 feet

83. A fly is in the center of the floor of a room wide, and 12 feet high. How far will it travel by the shortest path to one of the upper corners of the ceiling? Ans. V709+ft.

84. A corn crib 25 feet long holds 125 bushels. How many bushels will one of like shape and 35 feet long hold?

85. Let a cube be inscribed in a sphere, a second sphere in this cube, a second cube in this sphere, and so on; find the diameter of the 7th sphere, if that of the first is 27 inches. (2). What is the volume of all the spheres so inscribed including the first? Ans.

86. The area of a rectangular building lot is 720 sq. ft; its sides are as 4 to 5; what will it cost to excavate the earth 7 feet deep at 36¢ per cubic yard? Ans. $67.20.

87. A owns and B the remainder of a field 60 rods long and 30 rods wide at one end and 20 rods wide at the other end, both ends being parallel to the same side of the field They propose to lay out through it, parallel with the ends, a road one rod wide Isaving A's of the remainder at the wide end and B's 2% at the narrow end of the field. Required the location and area of the road. Ans

How much grass will

88. The diameter of a circular field is 240 rods. be left after 7 horses have eaten all they can reach, the ropes which are allowed them being of equal lengths and attached to posts so located that each can touch his neighbor's territory and none can reach beyond the boundary of the field? Ans. 62.831853 A.

89. What is the diameter of a circle inclosing three equal tangent circles, if the area inclosed by the three equal circles is 1 acre? Ans.

The problem, To draw a third straight line through the inaccessable point of intersection of two (converging) straight lines, is both metrical and descriptive, that is to say, the required line may be found either by metrical or descriptive geometry, but the method by Descriptive Geometry is far the simpler.

The following are the solutions by both methods: