20. How many cords of wood in a pile 18 feet long, 6 feet high, and 4 feet wide? Ans. 3 cords. 21. How many cords in a pile 10 feet long, 5 feet high, 7 feet wide? Ans. 2 cords, 94 cubic feet. 22. How many cords in a pile 35 feet long, 4 feet wide, 4 feet high? Ans. 48 cords. 23. How many cords in a pile that measures 8 feet on each side ? Ans. 4 cords. 24. How many cords in a pile that is 10 feet on each side? Ans. 71 cords. NOTE. When wood is "corded in a pile 4 feet wide, by multiply ing its length by its height, and dividing the product by 4, the quotient is the cord-feet; and if a load of wood be 8 feet long, and its height be multiplied by its width, and the product divided by 2, the quotient is the cord-feet. 25. How many cords of wood in a pile 4 feet wide, 70 feet 6 inches long, and 5 feet 3 inches high? Ans. 11% cords. NOTE. Small fractions are rejected. 26. How many cords in a pile of wood 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high? Ans. 10177 cords. 27. Required the number of cords of wood in a pile 100 feet long, 4 feet wide, and 6 feet 11 inches high. Ans. 218. 28. Agreed with a man for 10 cords of wood, at $5.00 a cord; it was to be cut 4 feet long, but by mistake it was cut only 46 inches long. How much in justice should be deducted from the stipulated price? Ans. $2.08. 29. If a load of wood be 8 feet long, 3 feet 8 inches wide, and 5 feet high, how much does it contain? Ans. 94 cord-feet. 30. If a load of wood be 8 feet long, 3 feet 10 inches wide, and 6 feet 6 inches high, how much does it contain ? Ans. 12 cord-feet. 31. If a load of wood be 8 feet long, 3 feet 6 inches wide, how high should it be to contain 1 cord? Ans. 4ft. 6 102". 32. If a load of wood be 12 feet long, and 3 feet 9 inches wide, how high should it be to contain 2 cords? Ans. 5ft. 8' 31". 33. D. H. Sanborn's parlour is 17ft. 9in. long, 14ft. 8in. wide, and 8ft. 9in. high. There are two doors 3ft. 4in. wide, and 7ft. high, and four windows 5ft. 3in. high, and 3ft. 4in. wide; the mop-boards are 9in. high. B. Gordon, a first-rate mason, will charge 10 cents per square yard for plastering the room. The paper for the room is 20 inches wide, and costs 6 cents per yard. E. Eaton will "paper "the room for 4 cents per square yard. Each window has 12 lights of 10in. by 14in. glass, the price of which is 124 cents per square foot. The painter's bill for setting the glass is 8 cents per light, and for painting the floor, mop-boards, and doors is 25 cents per square yard. What is the amount of Mr. Sanborn's bill? Ans. $33.72 89 SECTION LX. INVOLUTION. INVOLUTION is the raising of powers from any given number, as a root. A power is a quantity produced by multiplying any given number, called a root, a certain number of times continually by itself; thus, 2 2 is the root, or 1st power of 2 = 21. 2=22. 2=23. 2 x 2 = 2 × 2 × 2 = 2×2×2×216 is the 4th power, or biquadrate of 2 = 24. The number denoting the power is called the index or exponent of the power. Thus, the fourth power of 3, = 81, is expressed by 34, and 4 is the index or exponent; and the second power of 7, 49, is expressed by 72. = To raise a number to any power required. RULE.-Multiply the given number continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required. EXAMPLES. 1. What is the 5th power of 4? 4x4x4x4 x4 = 1024 Ans. 2. What is the 3d power of 8? Ans. 512. Ans. 282475249. 8. What is the 6th power of 13 ? 9. What is the 4th power of .045? 10. What is the 0 power of 1728? EVOLUTION, Ans. 1612. Ans. .000004100625. Ans. 1. OR THE EXTRACTION OF ROOTS. EVOLUTION is the reverse of Involution, and teaches to find the roots of any given powers. The root is a number whose continual multiplication into itself produces the power, which is denominated the 2d, 3d, 4th, &c., power, according to the number of times which the root is multiplied into itself. Thus, 4 is the square root of 16, because 4 x 4 = 16; and 3 is the cube root of 27, because 3 × 3 × 3 =27; and so on. = Although there is no number of which we cannot find any power exactly, yet there are many numbers of which precise roots can never be determined; but, by the help of decimals, we can approximate towards the root to any assigned degree of exactness. The roots which approximate are called surd roots; and those which are perfectly accurate are called rational roots. 3 Roots are sometimes denoted by writing the character ✔ before the power, with the index of the root over it; thus, the 3d root of 36 is expressed 36, and the second root of 36 is ✔ 36, the index 2 being omitted when the square root is designed. If the power be expressed by several numbers with the sign + or between them, a line is drawn from the top of the sign over all the parts of it; thus, the 3d root of 42+22 is √42+22, and the second root of 59 - 17 is 59 - 17, &c. Sometimes roots are designated like powers with fractional indices. Thus the square root of 15 is 152, the cube root of 21 is 21, and the 4th root of 37-20 is 37-20, &c. It sometimes will happen that one root is involved in another, thus: 125—5+19 +6, or 161/147. √ √ 178+7 √33—8+ŵ84 — 5 — √/87 + 16. SECTION LXI. EXTRACTION OF THE SQUARE ROOT. 1. LET it be required to find what number multiplied into itself will produce 1296. OPERATION, In contemplating this 1296(30+ 6 = 36 Ans. problem, we perceive 900 60666)396 396 that the root or number sought must consist of two figures, since the product of any two numbers can have at most but as many figures as there are in both factors, and at least but one less. We perceive, also, that the first figure of the root multiplied by itself must give a number not exceeding 12, and as 12 is not the second power of any number, and the second power of 4 is more than 12, we take 3 for the first figure of the root, which multiplied into itself gives 9. Now, the second power of 3, considered as occupying the place of tens, is 900, of which we have the root 30. Taking 900 from 1296, we have a remainder, 396; and having found the root of 900, we are now to seek a number, which, being added to this root (30) and multiplied into itself once, and into 30 twice, will produce 396. This number is found by dividing 396 by twice 30 plus the number sought. Q. E. D. * NOTE. Owing to the fact that the number of figures in the product of any two numbers is always limited as above stated, we ascertain the number of figures in the root of any given second power by putting a dot over the place of units, then over the place of hundreds, and so on. The number of dots gives the number of figures in the root. Thus the square root of 133225 consists of three figures. 2. What is the square root of 576 ? OPERATION. To illustrate this question in a different 576(24 Ans. way from the first, we will suppose that we 400 44)176 176 * have 576 tiles, each of which is one foot square, and we wish to know the side of a square room whose floor they will pave or cover. By adding 6 to 30 and multiplying the sum (36) into itself, we can easily see that we multiply 6 by itself once, and 30 by 6 twice, since 30 is contained in both factors, and in the operation is multiplied by the 6 in each. |