Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before; and so on, till the whole is completed. NOTE 1. - In separating the given number into periods, when the number of the figures is not divisible exactly by 3, the left-hand period will contain less than 3 figures. NOTE 2. The observations made in Notes 2, 3, and 4, under square root, are equally applicable to the cube root, except in pointing off decimals each period must contain three figures, and two ciphers must be placed at the right of the divisor when it is not contained in the dividend. EXAMPLES FOR PRACTICE. 1. What is the cube root of 78402752? Ans. 428. Ans. 42. Ans. 57. Ans. 432. Ans. 561. Ans. 636. Ans. 123. 2. What is the cube root of 74088 ? 3. What is the cube root of 185193? 4. What is the cube root of 80621568? 5. What is the cube root of 176558481? 6. What is the cube root of 257259456? 7. What is the cube root of 1860867? 8. What is the cube root of 1879080904 ? 9. What is the cube root of 41673648.563? Ans. 346.7. 10. What is the cube root of 483921.516051? Ans. 78.51. 11. What is the cube root of 8.144865728 ? Ans. 2.012. 12. What is the cube root of .075686967? Ans. 1234. Ans. .423. QUESTION. How many ciphers must be placed at the right of the divisor when it is not contained in the dividend? ART. 296. When it is required to extract the cube root of a common fraction, or a mixed number, it is prepared in the same manner as directed in square root. (Art. 272.) EXAMPLES FOR PRACTICE. 1. What is the cube root of 815 ? 2. What is the cube root of 729 ? 4096 3. What is the cube root of 49,87 ? APPLICATION OF THE CUBE ROOT. ART. 297. THE cube root may be applied in finding the dimensions and contents of cubes and other solids. 1. A carpenter wishes to make a cubical cistern that shall contain 2744 cubic feet of water; what must be the length of one of its sides? Ans. 14 feet. 2. A farmer has a cubical box that will hold 400 bushels of grain; what is the height of the box? Ans. 7.92 feet. 3. There is a cellar, the length of which is 18 feet, the width 15 feet, and the depth 10 feet; what would be the depth of another cellar of the same size, having the length, width, and depth equal? Ans. 13.92 feet. ART. 298. A SPHERE is a solid bounded by one continued convex surface, every part of which is equally distant from a point within, called the centre. The diameter of a sphere is a straight line passing through the centre, and terminated by the surface; as A B. A B ART. 299. A CONE is a solid having a circle for its base, and its top terminated in a point, called the vertex. QUESTIONS. - Art. 296. How is a common fraction or a mixed number prepared for extracting the square root?- Art. 297. To what may the cube root be applied? - Art. 298. What is a sphere? What is the diameter of a sphere? Art. 299. What is a cone? The altitude of a cone is its perpendicular height, or a line drawn from the vertex perpendicular to the plane of the base; as B C. ART. 300. Spheres are to each other as the cubes of their diameters, or of their circumferences. Similar cones are to each other as the cubes of their altitudes, or the diameters of their bases. All similar solids are to each other as the cubes of their homologous or corresponding sides, or of their diameters. ART. 301. To find the contents of any solid which is similar to a given solid. RULE. State the question as in Proportion, and cube the given sides, diameters, altitudes, or circumferences, and the fourth term of the proportion is the required answer. ART. 302. To find the side, diameter, circumference, or altitude, of any solid, which is similar to a given solid. RULE. State the question as in Proportion, and cube the given sides, diameters, circumferences, or altitudes, and the cube root of the fourth term of the proportion is the required answer. EXAMPLES FOR PRACTICE. 1. If a cone 2 feet in height contains 456 cubic feet, what are the contents of a similar cone, the altitude of which is 3 feet? Ans. 1539 cubic feet. OPERATION, 23:33:456:1539. 2. If a cubic piece of metal, the side of which is 2 feet, is worth $6.25, what is another cubical piece of the same kind worth, one side of which is 12 feet? Ans. $1350. 3. If a ball, 4 inches in diameter, weighs 50lb., what is the weight of a ball 6 inches in diameter ? Ans. 168.7+lb. QUESTIONS. What is the altitude of a cone? Art. 300. What proportion do spheres have to each other? What proportion do cones have to each other? What proportion do all similar solids have to each other? — Art. 301. What is the rule for finding the contents of a solid similar to a given solid? Art. 302. What is the rule for finding the side, diameter, &c., of a solid similar to a given solid? 1 4. If a sugar loaf, which is 12 inches in height, weighs 161b., many inches may be broken from the base, that the residue Ans. 2.5 in. how may weigh 8lb. ? 5. If an ox, that weighs 800lb., girts 6 feet, what is the weight of an ox that girts 7 feet? Ans. 1270.3lb. 6. If a tree, that is one foot in diameter, make one cord, how many cords are there in a similar tree, whose diameter is two feet? Ans. 8 cords. 7. If a bell, 30 inches high, weighs 1000lb., what is the weight of a bell 40 inches high? Ans. 2370.3lb. 8. If an apple, 6 inches in circumference, weighs 16 ounces, what is the weight of an apple 12 inches in circumference? Ans. 128 ounces. 9. A and B own a stack of hay in a conical form. It is 15 feet high, and A owns of the stack; it is required to know how many feet he must take from the top of it for his share. Ans. 13.1+ feet. § XXXIX. ARITHMETICAL PROGRESSION. ART. 303. WHEN a series of numbers increases or decreases by a constant difference, it is called Arithmetical Progression, or Progression by Difference. Thus, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. 29, 26, 23, 20, 17, 14, 11, 8, 5, 2. The first is called an ascending series or progression. The second is called a descending series or progression. The numbers which form the series are called the terms of the progression. The first and last terms are called the extremes, and the other terms the means. The constant difference is called the common difference of the progression. Any three of the five following things being given, the other two may be found: QUESTIONS. Art. 303. What is arithmetical progression? What is an ascending series? What a descending series? What are the terms of a progression? What the extremes? What the means? ART. 304. To find the common difference, the first term, last term, and number of terms, being given. ILLUSTRATION. In the following series, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 2 and 29 are the extremes, 3 the common difference, 10 the number of terms, and the sum of the series 155. It is evident that the number of common differences in any series must be 1 less than the number of terms. Therefore, since the number of terms in this series is 10, the number of common differences will be 9, and their sum will be equal to the difference of the extremes; hence, if the difference of the extremes (29-2=27) be divided by the number of common differences, the quotient will be the common difference. Thus, 27 ÷ 9= 3, the common difference. Hence the following RULE. - Divide the difference of the extremes by the number of term less one, and the quotient is the common difference. EXAMPLES FOR PRACTICE. 1. The extremes of a series are 3 and 35, and the number of terms is 9; what is the common difference? Ans. 4. 2. If the first term is 7, the last term 55, and the number of terms 17, required the common difference. Ans. 3. 3. If the first term is 4, the last term 14, and the number of terms 15, what is the common difference? Ans.. 4. If a man travels 10 days, and the first day goes 9 miles, QUESTIONS. Art. 304. What is the common difference? What five things are named, any three of which being given the other two can be found? What is the rule for finding the common difference, the first term, last term, and number of terms, being given? |