north 80 miles, the other due west 60 miles. How far apart are they? 5. What is the side of a square whose area is 121 square feet. 6. What is the distance between two opposite corners of a lot 50 feet by 50 feet? 7. What is the distance between two square whose area is 900 square feet? 8. What is the distance between two rectangle 15 rods long by 20 rods wide? Ans. 70.7 ft.+ opposite corners of a Ans. 42.426 ft. + opposite corners of a 9. What distance will I save by walking directly across, from one corner of a plantation a mile square to the opposite corner, in stead of following the two sides? Ans. 187.452 rd. 10. A person lays out two circular plots, one containing 9 times as much land as the other. How do their diameters compare? Cube Root. 513. Extracting the cube root of a number is resolving it into three equal factors; as, 8 = 2 × 2 × 2. 514. Taking the smallest and the greatest number that can be expressed by one figure, by two and three figures, let us see how the number of figures they contain compares with the number of figures in the cubes :— Roots, Cubes, 11 10 99 100 999 9 729 1'000 970'299 1'000'000 997'002'999 We find from these examples that, if we separate a cube into periods of three figures each, commencing at the right, there will be as many figures in the cube root as there are periods in the cube,-counting the left-hand figure or figures, if there are but one or two, as a period. 515. We derive the method of extracting the cube root from the opposite operation of cubing. Cube 25, regarding it as composed of 2 tens (20) and 5 units. 513. What is meant by extracting the cube root of a number?-514. How can we find, from a cube, the number of figures its cube root contains ?-515. Whence do we derive the method of extracting the cube root? Cube 20 + 5. The square of 20 + Multiplying by 20, 5 was found in § 501; we multiply it by 20 + 8. 202+2 (20 × 5) + 52 Adding partial prod's, 203 +3 (202 × 5) + 3 (20 × 52) + 53 = 253 As 5 is a common factor of the last three terms, the cube of 25, as just found, may be written as follows: 3 times the square of the tens cube of the tens + + 3 times product of tens and units the units. + the square of the units 516. Reverse the process; find the cube root of 15625. According to § 514, we separate 15625 into periods of three figures each, beginning at the right (15'625), and find that the root will contain two figures,-a tens' and a units' figure. The cube of the tens must be found in the left-hand period 15(000). The greatest number whose cube is contained in 15(000) is 2(0), which we place on the right as the tens' figure of the root. 2 tens (20) cubed 8 thousands, which we subtract from the 15 thousands. Bringing down the remaining period, we have 7625; which, § 515, must equal 15'625 (2 8 7625 3 times the square of the tens + 3 times the product of the tens and units x the units. + the square of the units Hence, to find the units' figure of the root, we divide 7625 by 3 times the square of the tens as a trial divisor. It is contained 6 times; but, making allowance for the completion of the trial divisor, we 15'625 (25 8 regard the quotient as 5, and write 5 in the root as its units' Trial div., 202 × 3 = 1200 7625 figure. Now, to complete the divisor, we have to add to 3 times the square of the tens, al 20 × 5 × 3 = 300 Complete divisor, 1525 7625 ready found, 3 times the product of the tens and units (20 × 5 × 3 = 300), and the square of the units (52 = 25),--making 1525. Multiplying this by the units' figure, and subtracting, we have no remainder. Ans. 25. How may the cube just found be written? Hence, what does the cube of a number composed of tens and units equal?-516. Reverse the process; extract the eube root of 15625, explaining the steps.-517. Recite the rule. 517. RULE.-1. Separate the given number into periods of three figures each, beginning at the units' place. 2. Find the greatest number whose cube is contained in the left-hand period, and place it on the right as the first root figure. Subtract its cube from the first period, and to the remainder annex the second period for a dividend. 3. Take three times the square of the root already found; and, annexing two ciphers, place it on the left as a trial divisor. Find how many times the trial divisor is contained in the dividend (making some allowance), and annex the quotient to the root already found. Complete the trial divisor, by adding to it 30 times the product of the last root figure and the root previously found, also the square of the last root figure. Multiply the divisor, thus completed, by the last root figure, subtract the product from the dividend, and bring down the next period as before. 4. Repeat the processes in the last paragraph, till the periods are exhausted. If any trial divisor is not contained in its dividend, place 0 in the root, annex two ciphers to the trial divisor, bring down the next period, and find how many times it is then contained. If, on multiplying a completed divisor by the last root figure, the product is greater than the dividend, the last root figure must be diminished, and the necessary changes made in completing the divisor. Separate a decimal into periods, from the decimal point to the right, completing the last period, if necessary, by annexing one or two ciphers. To find the cube root of a common fraction, see § 505. To prove the operation, cube the root found. Ex. 2.-Extract the cube root of 348616.378872. 1. Extract the cube root of 2357947691. Ans. 1331. 2. What is the cube root of 91125? Of 7256313856? Of 387420489? Of 10077696 ? Sum of ans. 2926. 3. Extract the cube root of 42875. Of 125450540216. Of 343558903294872. Of 117649. 4. What is the cube root of 18.609625? Sum of ans. 75128. Of.065450827? Of .000000008? Of 1.25992105, carried to five decimal places? Of 3, to four decimal places? 5. Find the cube root of . 62 Sum of ans. 5.57725. 32 Of 3845. Of 171. Of 18. Of 3458. Of. Of. Of 1014. Ans. 1, 13, 2.577+, &c. 686 518. The solid contents of similar bodies are to each other as the cubes of their like dimensions. The solid contents of two globes whose diameters are 6 in. and 12 in., are to each other as 63 to 123, or 216 to 1728: 519. When the solid contents of a cube are known, extract the cube root, to find one of the sides. The answer will be in the denomination of linear measure that corresponds to the denomination of the solid contents. A cubical block whose solid contents are 8 cubic inches, will be 2 (38) linear inches on each side. 6. If a ball 3 in. in diameter weighs 8 lb., what will a ball of equal density, whose diameter is 4 in., weigh? Ans. 1884 lb. 7. What is the side of a cube whose solid contents equal those of a rectangle, 8 ft. 3 in. long, 3 ft. wide, and 2 ft. 7 in. deep? Ans. 47.9843 in. 8. What is the side of a cube containing 2197 cu. in.? 9. There are three balls whose diameters are respectively 3, 4, and 5 inches. What is the diameter of a fourth ball, of the same density, equal in weight to the three? Ans. 6 in. 10. If a ball 12 in. in diameter weighs 238 lb., what will be the diameter of another ball of the same metal, weighing 32 lb. ? 518. What principle is laid down respecting the solid contents of similar bodies? -519. How is the side of a cube found from its solid contents ? CHAPTER XXXVII. PROGRESSION. 520. Progression is a regular increase or decrease in a series of numbers. 521. There are two kinds of Progression, Arithmetical and Geometrical. A series of numbers are said to be in Arithmetical Progression, when they increase or decrease by a common difference: as, 16, 18, 20, 22; 16, 14, 12, 10. A series of numbers are said to be in Geometrical Progression, when they increase or decrease by a common ratio: as, 16, 32, 64, 126; 16, 8, 4, 2. 522. The numbers forming the series are called Terms. The first and the last term are the Extremes, the intermediate terms the Means. 523. When the terms increase, they form an Ascending Series; when they decrease, a Descending Series. Arithmetical Progression. 524. In Arithmetical Progression, there are five things to be considered: the First Term, the Last Term, the Number of Terms, the Common Difference, and the Sum of the Series. Three of these being given, the other two can be found. To find the relations between these five elements, let us look at the series that follow, in which the first term is 13, the common difference 2, and the number of terms 5:- Ascending, 13, 13+2, 13+2+2, 13+2+2+2, 13+2+2+2+2. Descending, 13, 13-2, 13-2-2, 13-2-2-2, 13-2-2-2-2. It will be seen that the second term equals the 1st, plus (in the descending series, minus) once the common difference; the third term 520. What is Progression ?-521. How many kinds of Progression are there? When are numbers said to be in Arithmetical Progression? When, in Geometrical Progression? Give examples.-522. What are the numbers forming the series called? What are the Extremes? What are the Means?-523. What is an Ascending Series? What is a Descending Series?-524. How many things are to be con. sidered in Arithmetical Progression? Name them. How many of these must be given, to find the rest? |