3. What is the square root of 43264? 7. What is the square root of '001296? Ans. 208. Ans. 999. Ans. 15'3. Ans. 31'05671. Ans. '54. Ans. 6031. Ans. 12'8+. NOTE 2. In the last example, there was a remainder, after all the figures were brought down. In such cases, the precise root can never be obtained. For, as the operation is continued by annexing ciphers, the last figure of every dividend must be a cipher. But the root figure obtained from this dividend, is also placed at the right hand of the divisor, and consequently is multiplied into itself, and the last figure of the product placed under the cipher, which is the last figure of the dividend, to be subtracted from it. And as the product of no one of the significant figures ends in a cipher, there will always be a remainder. 11. What is the square root of 3? Ans. 1'73+. Ans. 3'16. Ans. 13'57. NOTE 3.Since, from the rule for multiplying one fraction by another, a fraction is involved by involving its numerator and its denominator, the root of a fraction is obtained by finding the root of its numerator, and of its denominator. 15. What is the square root of? 16. What is the square root of 16? 17. What is the square root of ? 18. What is the square root of 201? NOTE 4. Ans. . Ans.. When the numerator and denominator are not exact squares, the fraction may be reduced to a decimal, and the approximate root found. 19. What is the square root of='75? 20. What is the square root of 25. Ans. '866+. Ans. '912+ PRACTICAL EXERCISES IN THE EXTRACTION OF T 210. THE SQUARE ROOT. Ans. 64. 1. A general has 4096 men; how many must he place in rank and file to form them into a square? 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 large? 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 1.-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 2. A square 4 inches on one side, contains 16 square inches; one twice as long, or 8 inches on each side, contains 64 square inches, 4 times 16; one 3 times as long, or 12 inches on each side, contains 1449 times 16 square inches. It may also be shown by geometry, that if the diameter of a circle be doubled, its contents will be increased 4 times; if the diameter be trebled, the contents will be increased 9 times. That is, the contents of squares are in proportion to the squares of their sides, and the contents of circles are in proportion to the squares of their diameters. Hence, to perform the above example, square the diameter, multiply the square by 9, and extract the square root of the product. Ans. 12 inches. 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. 1732 feet. 13. If the diameter of a circle be 12 inches, what is the diameter of one as large? Ans. 6 inches. 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? B A Fig. 1. NOTE 3. 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. It is proved by a geometrical demonstration that the square contents of a square formed on the longest side, A C, are equal to the square contents of the two squares, one formed on each of the other two sides, A B, and C B. Thus, Fig. 2, a square formed on A B, the shortest side, will contain 9 square feet, the square on C B will contain 16 square feet, 9+16=25 square feet, in both squares. The square on A C contains 25 small squares of the same size as the squares on the other two sides are divided into, or 25 square feet, and the square root of 25 will be the length of the longest side, or, Ans., 5 feet. Hence, if the length of the two short sides are given, square each, add the squares together, and extract the square square root of the sum; the root will be the length of the long side. If the long side, and one of the short sides are given, square each, subtract the square of the short side from the square of the long side; the square root of the remainder will be the other short side. EXAMPLES. 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? 18. The ladder and ditch given, required the wall. Ans. 24 feet. Ans. 32 feet. 19. The distance between the lower ends of two equal rafters is 32 feet, and the hight 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 feet. ft.; and 22'36+ 21. There is a field 800 rods long and 600 rods wide; what is the distance between two opposite corners? Ans. 1000 rods. how many 22. There is a square field containing 90 acres ; rods Questions. 210. How does it affect the contents of a square to double its length? to treble its length? How does it affect the contents of circles to double or treble their diameters? How will you find the diameter of a circle nine times as large as one of a given diameter? What is a right angled triangle? What is said of the squares on its sides? How shown by Fig. 2? When both short sides are given, how do you find the long side? When the long side, and one short side are given, how do you find the other? 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 Root. T211. 1. How many feet in length is each side of a cubic block, containing 125 solid feet? SOLUTION. - As the solid contents of a cubical body are found, when one side is known, by involving the side to the third power, or cube, (T 206,) so when the solid contents are known, we find the length of one side by extracting the cube root, a number, which, taken as a factor 3 times, will produce the given number, (T 207.) The cube root of 125 we find by inspection, or by the table, 206, to be 5. Ans. 5 feet. 216 solid feet? 2. What is the side of a cubic block, containing 64 solid feet? 27 solid feet? 512 solid feet? Answers, 4 ft., 3 ft., 6 ft., and 8 ft. 3. 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? 20 D SOLUTION.It is evident that, as in the former examples, we must find the length of one side of a cubical pile which 13824 such blocks will make by extracting the cube root of 13824. But this number is so large, that we cannot so easily find the root as in the former examples; we will endeavor, however, to do it by a sort of trial; and, 1st. We will try to ascertain the number of figures, of which the root 20 will consist. This we may do by pointing the number off into periods of three figures each. For the cube of any figure will contain 3 times as many, or 1 or 2 less than 3 times as many figures as the number itself. The cube of 2 contains 1 figure; the cube of 5 contains 2 figures; the cube of 9 contains 3 figures; the cube of 10 contains 4 figures, and so on. Pointing off, we see that 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 inspection, 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. As the root is one side of a cube, let us form a cube, (Fig. I.,) each side of which shall be regarded 20 feet, expressed by the root now obtained. The contents of this cube are 20 X 20 X 20-8000 solid feet, 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 Now as each side is 20 feet square, its square contents are 400 square feet, and the square contents of the 3 sides are 1200 square feet. Hence, an addition of 1 foot thick would require 1200 solid feet, and C. dividing 5824 solid feet by 1200 solid feet, the contents of the addition 1 foot thick, and we get the thickness of the addition. It will be seen that the quotient figure must not always be as large as it can be. There might be enough, for instance, to make the three additions now under consideration 5 feet thick, when there would not then be enough remaining to complete the additions. The divisor, 1200, is contained 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 1 4800, is the solid feet contained in these additions; but there are still 1024 feet left, and 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 20; 16 X 20 = = |