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prove the work? 17. Why do we point off mixed numbers both ways from units? 18. When there is a remainder, how may we continue the operation? 19. Why can we never obtain the precise root of surd numbers? 20. How do we extract the square root of vulgar fractions?
1. A general has 4096 men; how many must he place in rank and file to form them into a square? Ans. 64. 2. If a square field contains 2025 square rods, how many rods does it measure on each side? Ans. 45 rods. 3. How many trees in each row of a square orchard containing 5625 trees? Ans. 75. 4. There is a circle, whose area, or superficial contents, is 5184 feet; what will be the length of the side of a square of equal area? 518472 feet, Ans.
5. A has two fields, one containing 40 acres, and the other containing 50 acres, for which B offers him a square field containing the same number of acres as both of these; how many rods must each side of this field measure?
Ans. 120 rods.
6. If a certain square field measure 20 rods on each side, how much will the side of a square field measure, containing 4 times as much? 20 X 20 X 4: =40 rods, Ans. 7. If the side of a square be 5 feet, what will be the side of one 4 times as large? 9 times as large? 16 times as large? 25 times as large? 36 times as iarge? Answers, 10 ft.; 15 ft.; 20 ft.; 25 ft.; and 30 ft. 8. It is required to lay out 288 rods of land in the form of a parallelogram, which shall be twice as many rods in length as it is in width.
Note. If the field be divided in the middle, it will form two equal squares.
Ans. 24 rods long, and 12 rods wide. 9. I would set out, at equal distances, 784 apple trees, so that my orchard may be 4 times as long as it is broad; how many rows of trees must I have, and how many trees in each row? Ans. 14 rows, and 56 trees in each row. 10. There is an oblong piece of land, containing 192 square rods, of which the width is as much as the length; required its dimensions. Ans. 16 by 12.
11. There is a circle, whose diameter is 4 inches; what is the diameter of a circle 9 times as large?
Note. The areas or contents of circles are in proportion to the squares of their diameters, or of their circumferences. Therefore, to find the diameter required, square the given diameter, multiply the square by the given ratio, and the square root of the product will be the diameter required. No4 × 4 × 9 = 12 inches, Ans. 12. There are two circular ponds in a gentleman's pleasure ground; the diameter of the less is 100 feet, and the greater is 3 times as large; what is its diameter? Ans. 173'2+ feet 13. If the diameter of a circle be 12 inches, what is the diameter of one as large ? Ans. 6 inches.
¶ 109. 14. A carpenter has a large wooden square; one part of it is 4 feet long, and the other part 3 feet long; what is the length of a pole, which will just reach from one end to the other?
Note. A figure of 3 sides is called a triangle, and, if one of the corners be a square corner, or right angle, like the angle at B in the annexed figure, it is called a right-angled triangle, of which the square of the longest side, A C, (called the hypotenuse,)
is equal to the sum of the squares of the other two sides, A B
and B C.
16, and 32 = 9; then, /9+ 16 = 5 feet, Ans. 15. If, from the corner of a square room, 6 feet be measured off one way, and 8 feet the other way, along the sides of the room, what will be the length of a pole reaching from point to point? Ans. 10 feet.
16. A wall is 32 feet high, and a ditch before it is 24 feet wide; what is the length of a ladder that will reach from the top of the wall to the opposite side of the ditch?
Ans. 40 feet.
17. If the ladder be 40 feet, and the wall 32 feet, what is the width of the ditch? Ans. 24 feet.
18. The ladder and ditch given, required the wall.
Ans. 82 feet
19. The distance between the lower ends of two equal rafters is 32 feet, and the height of the ridge, above the beam on which they stand, is 12 feet; required the length of each rafter. Ans. 20 feet.
20. There is a building 30 feet in length and 22 feet in width, and the eaves project beyond the wall 1 foot on every side; the roof terminates in a point at the centre of the building, and is there supported by a post, the top of which is 10 feet above the beams on which the rafters rest; what is the distance from the foot of the post to the corners of the eaves? and what is the length of a rafter reaching to the middle of one side? a rafter reaching to the middle of one end? and a rafter reaching to the corners of the eaves? Answers, in order, 20 ft.; 15'62 ft.; 18'86 + ft.; and 22'36+ ft.
21. There is a field 800 rods long and 600 rods wide; what is the distance between two opposite corners ?
Ans. 1000 rods.
22. There is a square field containing 90 acres ; how many rods in length is each side of the field? and how many rods apart are the opposite corners ?
Answers, 120 rods; and 169'7+rods.
23. There is a square field containing 10 acres; what distance is the centre from each corner? Ans. 28'28+ rods.
EXTRACTION OF THE CUBE
¶ 110. A solid body, having six equal sides, and each of the sides an exact square, is a CUBE, and the measure in length of one of its sides is the root of that cube; for the length, breadth and thickness of such a body are all alike; consequently, the length of one side, raised to the 3d power, gives the solid contents. (See T 36.)
Hence it follows, that extracting the cube root of any number of feet is finding the length of one side of a cubic body, of which the whole contents will be equal to the given uumber of feet.
1. What are the solid contents of a cubic block, of which each side measures 2 feet? Ans. 23 2 X 2 X 28 feet. 2. How many solid feet in a cubic block, measuring 5 feet on each side? Ans. 53 = = 125 feet.
3. How many feet in length is each side of a cubic block, containing 125 solid feet? Ans. 1255 feet.
Note. The root may be found by trial.
4. What is the side of a cubic block, containing 64 solid feet? 27 solid feet? 216 solid feet? 512 solid Answers, 4 ft.; 3 ft.; 6 ft.; and 8 ft. 5. Supposing a man has 13824 feet of timber, in separate blocks of 1 cubic foot each; he wishes to pile them up in a cubic pile; what will be the length of each side of such a pile?
It is evident, the answer is found by extracting the cube root of 13824; but this number is so large, that we cannot so easily find the root by trial as in the former examples ;We will endeavour, however, to do it by a sort of trial; and, 1st. We will try to ascertain the number of figures, of which the root will consist. This we may do by pointing the number off into periods of three figures each (T 107, ex. 1.)
Pointing off, we see, the root will consist of two figures, a ten and a unit. Let us, then, seek for the first figure, or tens of the root, which must be extracted from the left hand period, 13, (thousands.) The greatest cube in 13 (thousands) we find by trial, or by the table of powers, to be 8, (thousands,) the root of which is 2, (tens;) therefore, we place 2 (tens) in the root. The root, it will be recollected, is one side of a cube. Let us, then, form a cube, (Fig. I.) each side of which shall be supposed 20 feet, expressed by the root now obtained. The contents of this cube are 20X20X20 8000 solid feet,
8000 feet, Contents. which are now disposed of, and which, consequently, are to be deducted from the whole number of feet, 13824. 8000 taken from 13824 leave. 5824 feet. This deduction is most readily performed by subtracting the cubic number, 8, or the cube of 2, (the figure of the root already found,) from
the period 13, (thousands,) and bringing down the next period by the side of the remainder, making 5824, as before.
2d. The cubic pile A D is now to be enlarged by the addition of 5824 solid feet, and, in order to preserve the cubic form of the pile, the addition must be made on one half of its sides, that is, on 3 sides, a, b, and c. Now, if the 5824 solid feet be divided by the square contents of these 3 equal sides, that is, by 3 times, (20 X 20400) 1200, the quotient will be the thickness of the addition made to each of the sides a, b, c. But the root, 2, (tens,) already found, is the length of one of these sides; we therefore square the root, 2, (tens,) =20 X 20 400, for the square contents of one side, and multiply the product by 3, the number of sides, 400 × 3 = 1200; or, which is the same in effect, and more convenient in practice, we may square the 2, (ters) and mul tiply the product by 300, thus, 2 x2=4, and 4 × 300=1200, for the divisor, as before.
The divisor, 1200, is conOPERATION-CONTINUED. tained in the dividend 4 times; consequently, 4 feet is the thickness of the addition made to each of the three sides, a, b, c, and 4 X 1200 = 4800, is the solid feet contained in these additions; but, if we look at Fig. II., we shall perceive, that this addition to the 3 sides does not complete the cube; for there are deficiencies in the 3 corners n, n, n. Now the length of each of these deficiencies is the same as the length of each side, that is, 2 (tens) 20, and their width and thickness are each equal to the last quotient figure, (4); their contents, therefore, or the number of feet required to fill these deficiencies, will be found by multiplying the square of the last quotient figure, (42) =16, by the length of all the deficiencies, that is, by 3 times
13824 (24 Root.
Divisor, 1200)5824 Dividend.