179. SIMILAR SOLIDS.

42. There is a cubical box, each of whose inside dimensions is 1 inch. What is the content of the cube? What is the content of the largest globe that can be included within the box? Ans., .5236 of a cubic inch.

43. Suppose each side of the box to be 2 inches; what would be its capacity? How many times as large as the other? How times as large would be the globe that it would include?

44. Suppose each side of the box to be 3 inches. Ask and answer the same questions,

45. What is the content of a block of marble, if its length, width, and thickness are 5, 4, and 3 feet respectively? What, if 10, 8, and 6 feet? What, if 15, 12, and 9 feet? What, if 2, 2, and 1 feet?

NOTE 1. Similar solids are those whose corresponding sides or dimensions are proportional. Thus the different solids mentioned in example 45 are similar.

NOTE 2. The contents of similar solids are in proportion to the cubes of their corresponding dimensions; that is, The content of one solid is to the content of another similar solid, as the cube of any dimension of the former is to the cube of the like dimension of the latter.

46. Can you illustrate this truth by the above and similar examples?

47. Can you show from the above examples why the solidity of a sphere is obtained by multiplying the cube of the diameter by .5236?

NOTE 3. If a plane pass through a cone or pyramid parallel to its base, it divides the lines it meets proportionally; the small cone or pyramid cut off by it is, therefore, similar to the whole cone or pyramid.

48. If a ball 20 inches in diameter weighs 555 pounds, what is the diameter of one of the same metal that weighs 15 pounds?

49. If a vessel, one of whose sides is 2 feet, will contain 37.63 gallons, what will another similar vessel hold, whose corresponding side is 15 feet?

50. If a tree whose diameter is 2 feet at the base contains 3 cords of wood, how much wood will there be in a tree of the same shape, the diameter of which is 3 feet?

51. If an ox whose girt is 7 feet weighs 1000 pounds, how much will an ox of the same form weigh, whose girt is 6 ft. 6 in. ? What should be the girt of an ox of the same form, which weighs 1500 pounds?

52. A square pyramid of wood, 12 feet long, and each side of whose base is 18 inches, is to be balanced upon a pivot passing through it. How far from the base must the pivot be placed to balance it?

53. A cone is 15 feet high; how many feet of its top must be taken off to remove one half of it? one third of it? two thirds?

54. What part of the cone will a plane 5 feet from the base cut off?

55. If a stack of hay 12 feet high weighs 4 tons, how much will a similar stack weigh, that is 15 feet high?

56. If a common brick 8 inches long weighs 4 pounds, how much will a brick of similar shape weigh, that is 12 inches long?

57. If a man 6 feet high weighs 200 pounds, what will a giant of similar form and of equal solidity weigh, that is 81 feet high?

58. A cubic foot of lead weighs 11352 ounces; how much will a leaden ball 2 inches in diameter weigh? 4 inches? 6 inches? 8 inches? 12 inches?

59. How large a ball of lead will weigh 709 ounces?

NOTE. The ball will contain

709

11352

of a cubic foot.

✔.5236 answer. Ór, .5236: :: 13: the cube of the answer; that is, by Note 2, the solid content of a ball 1 foot in diameter, is to the solid content of the required ball, as the cube of the diameter of the first ball, is to the cube of the diameter of the second.

180. MENSURATION OF BOARDS AND TIMBER.

The unit of measure for boards, plank, joists, beams, &c., is the square foot; they are usually surveyed by board measure, the board being estimated at one inch thick. Thus, a board

10 ft. long, and 1 feet wide, contains 15 square feet, if it is 1 inch thick; if it is 11⁄2 inch thick, it contains 22 square feet; if 2 inches thick, 30 square feet. Round timber is sometimes measured by the ton, and sometimes by board measure.

RULE FOR MEASURING BOARDS, PLANK, JOIST, BEAMS, &C.

Multiply the length in feet by the width in inches, and this product by the depth, or thickness, in inches, and divide the last product by 12; the quotient will be the number of square feet.

1. How many square feet in a board 23 feet long, 17 inches wide, and 1 inch thick? in. thick? in. thick? 1 in. thick?

2. How many square feet in a joist 30 feet long, 6 in. wide, and 3 in. thick? 5 in. wide, and 21⁄2 in. thick?

To find the solid contents of any rectangular stick of timber that does not taper, see Art. 170; if it does taper, see Art. 174.

181. To find the side of the largest square stick of timber that can be hewn or sawn from a round log, whose diameter is given.

RULE. Multiply the diameter of the smaller end by .7071. (Art. 165, quest. 22.)

SECTION XX. —PROBLEMS IN MENSURATION.

From the principles and illustrations detailed in the two preceding sections, we deduce the following useful and practical problems.

182. To find the solid contents of the walls of a rectangular cistern or building, of any given dimensions.

RULE. From the OUTSIDE perimeter of the walls, (p. 214, def. 32,) subtract four times the thickness of the walls; the remainder will be the length of the walls. Then multiply this length by the height, and this product by the thickness.

183. To find the content of the gable ends.

RULE. Multiply the breadth of the house by the perpendicular height of the ridge above the eaves; the product will be the area of both gable ends. For the solid contents, Multiply this arca by the thickness.

184. To find the solid contents of the walls of a cylindrical shaped structure; as round cisterns, wells, &c.

RULE. To the inner diameter of the cylinder, add the thickness of the wall; the sum will be the mean of the inside and outside diameters. Multiply this sum by 34, or by 3.1416; the product will be the mean circumference. For the solid content, Multiply this mean circumference by the height, and this product by the thickness.

185. To find the solid contents of the bottom, or foundation-work, of a cylindrical cistern.

RULE Multiply the diameter from outside to outside by 34 for the circumference: then multiply half the diameter by half the circumference, and this product by the thickness, for the cubical content.

NOTE. To find the capacity of a cistern, the top and bottom of which are not equal, see Art. 174.

186. To find the number of bricks it will take to build any wall, or other work, the solid contents of which are known.

A common brick is 8 inches long, 4 inches wide, and 2 inches thick; its solid content is therefore 64 cubic inches, or of a cubic foot. Hence the

RULE. Multiply the number of cubic feet by 27; the product will be the answer.

187. To find how many gallons a rectangular cistern will contain.

RULE. From the inside dimensions find the cubical contents in feet, (170,) and multiply the content thus found by 7. This will be the number nearly. Or, Find the content in cubic inches, and divide by 231.

188. To find how many gallons a cylindrical cistern will contain.

RULE. Multiply the inside diameter by itself, and this product by the height, the dimensions being taken in feet; then multiply the last product by 53.

189. To find the number of quarts a cylindrical vessel will hold.

RULE. Take the dimensions in inches; multiply the square of the diameter by twice the height, and divide the product by 147.

190. To find the number of bushels à rectangular bin will hold.

RULE. Take the dimensions in feet; multiply the length, width, and height together; then multiply this product by 45, and divide by 56. Or, Take the dimensions in inches; multiply the length, width, height together; multiply this product by 5, and divide by 10752.

and

191. To find the inside dimensions of any box, cistern, &c., of a given capacity, the dimensions of which are to have a given proportion to each other.

RULE. Divide the capacity of the required box, &c., by the capacity of one whose dimensions are expressed by the numbers of the given proportion; and multiply each of these numbers by the cube root of the quotient; the several products will be the dimensions required.

1. Required the dimensions of a rectangular cistern which shall hold 3000 gallons, and whose length, width, and depth shall be as the numbers 5, 3, and 4.

SOLUTION.-5×3×4×7=450 gallons, the capacity of

a cistern whose dimensions are 5, 3 and 4 feet.

=

3000 ÷ 450

=

length;

-6.6 = 1.882. Then 1.882 × 59.41 ft. 1.882 35.646 ft. = the width; and 1.882 X 47.528 the height.

2. Required the inside dimensions of a round cistern to contain 10000 gallons, the diameter of which shall be to the height as 2 to 3.

SOLUTION.-2X2×3× 57 = 70 gallons, the capacity

of a cistern whose dimensions are 2 and 3 feet.

=141.8445.215. 5.215 × 2 = 10.43 ft. ter; 5.215 × 3 = 15.645 feet the height.

10000 ÷ 70

the diame

NOTE. Of rectangular forms, the cube gives the greatest capacity from a given amount of materials, if the top is to be covered; if not, a square base is the best, with an indefinite height.

In cylindrical forms, the greatest capacity from a given amount of materials is obtained by making the diameter and height equal, if the top is to be covered; if not, a given amount of materials will enclose the greatest space when the diameter is one half the height.