2. What is the cube root of $4965783 ? 3. What is the cube root of 436036824287. 4. What is the cube root of 84.604519 ? 5. Required the cube root of 54439939. 6. Extract the cube root of 60236288. 7. Extract the cube root of 109215352. 8. What is the cube root of 116.930169 ? 9. What is the cube root of .726572699 ? 10. Required the cube root of 2. 11. Find the cube root of 11. 12. What is the cube root of 122615327232 ? 13. What is the cube root of ? 1331? 14. What is the cube root of 13 15. What is the cube root of 12? 16. What is the cube root of 59319? 68921 Ans. 327. Ans. 7583. Ans. 379. Ans. 392. Ans. 478. Ans. 4.89. Ans. .899. Ans. 1.2599.+ Ans. 2.2239.Ans. 4968. Ans.. Ans. Ans.. Ans.. APPLICATION OF THE CUBE ROOT. Principles assumed. Spheres are to each other, as the cubes of their diameter. Cubes and all similar solid bodies, are to each other, as the cubes of their diameter, or homologus sides. 17. If a ball, 3 inches in diameter, weigh 4 pounds, what will be the weight of a ball that is 6 inches in diameter ? Ans. 32 lbs. 18. If a cube of gold 1 inch in diameter, is worth $120, what is the value of a globe 34 inches in diameter ? Ans. $5145. 19. If the weight of a well-proportioned man, 5 feet 10 inches in height, be 180 pounds, what must have been the weight of Goliath of Gath, who was 10 feet 4 inches in height? Ans. 1015.1.+lbs. 20. If a bell, 4 inches in height, 3 inches in width, and of an inch in thickness, weigh 2 pounds, what should be the dimensions of a bell that would weigh 2000 pounds? Ans. 3 ft. 4 in. high, 2 ft. 6 in. wide, and 2 in. thick. 21. Having a small stack of hay 5 feet in height weighing 1 cwt., I wish to know the weight of a similar stack, that is 20 feet in height. Ans. 64 cwt. 22. If a man dig a small square cellar, which will measure 6 feet each way, in one day, how long would it take him to dig a similar one that measured 10 feet each way ? 23. If an ox, whose girt is 6 feet, weighs weight of an ox whose girt is 8 feet? Ans. 4.629days. 600lbs., what is the Ans. 1422.2+lbs. A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. RULE. 1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs. 2. Find the first figure of the root by trial, or by inspection, in the table of powers, and subtract its power from the left hand period. 3. To the remainder, bring down the first figure in the next period, and call it the dividend. 4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power for a divisor. 5. Find how many times the divisor is contained in the dividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, and subtract it from the given number, as before. 7. Bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor, as be fore; and in like manner proceed till the whole is finished. 1. What is the cube root of 20346417 ? 3. What is the fifth root of 281950621875 ? 4. Required the sixth root of 1178420166015625. Ans. 325. 5. Required the seventh root of 1283918464548864. Ans. 195. Ans. 144. Required the eighth root of 218340105584896. Ans. 62. SECTION LV. ARITHMETICAL PROGRESSION.* WHEN a series of quantities, or numbers, increase or decrease by a constant difference, it is called arithmetical progression or progression by difference. The constant difference is called the common difference, or ratio of the progression. Thus, let there be the two following series, 1, 5, 9, 13, 17, 21, 25, 29, 33, 25, 22, 19, 16, 13, 10, 7, 4, 1. 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 progression. The first and last terms of progression, are called the extremes, and the other terms, the means. Any three of the five following things being given, the other two may be found. The first term, last term, and the number of terms being given, to find the common difference. RULE. Divide the difference of the extremes by the number of terms less one, and the quotient is the common difference. 1. The extremes are 3 and 45, and the number of terins is 22. What is the common difference? OPERATION. 45-3 -2 Answer. 22-1 * As a demonstration of this rule is so much better understood by Algebra than Arithmetic, the author has thought best to omit it. 2. A man is to travel from Boston to a certain place, in 11 days, and to go but 5 miles the first day, increasing every day by an equal increase, so that the last day's journey may be 45 miles. Required the daily increase. Ans. 4 miles. 3. A man had 10 sons, whose several ages differed alike; the youngest was 3 years, and the oldest 48. What was the common difference of their ages? Ans. 5 years. 4. A certain school consists of 19 scholars; the youngest is 3 years old, and the oldest 59. What is the common difference of their ages? Ans. 2 years. PROBLEM II. The first term, the last term, and the number of terms given to find the sum of all the terms. RULE. Multiply the sum of the extremes by the number of terms, and half the product will be the sum. 5. The extremes of an arithmetical series are 3 and 45, and the number of terms 22. Required the sum of the series. 6. A man going a journey, travelled the first day 7 miles, the last day 51 miles, and he continued his journey 12 days. How far did he travel? Ans. 348 miles. 7. In a certain school there are 19 scholars; the youngest is 3 years old, and the oldest 39. What is the sum of their ages? Ans. 399 years. 8. Suppose a number of stones were laid a rod distant from each other, for thirty miles, it being the distance from Boston to Haverhill; and the first stone a rod from a basket. What length of ground will that man travel over who gathers them up singly, returning with them one by one to the basket? Ans. 288090 miles, 2 rods. PROBLEM III. Given the extremes and the common difference, to find the number of terms. RULE. Divide the difference of the extremes by the common difference, and the quotient increased by one, will be the number of terms required 9. If the extremes are 3 and 45, and the common difference 2, what is the number of terms? OPERATION. 45-3 +1=22 Answer. 2 10. In a certain school where the ages of the scholars all differ alike; the oldest is 39 years old, and the youngest is 3 years, and the difference between the ages of each is 2 years. Required the number of scholars ? Ans. 19 scholars. 11. A man going a journey, travelled the first day 7 miles and the last day 51 miles, and each day increased his journey by 4 miles. How many days did he travel? PROBLEM IV. Ans. 12 days. The extremes and common difference given, to find the sum of all the series. RULE. Multiply the sum of the extremes by their difference, increased by the common difference, and the product divided by twice the common difference will give the sum. 12. If the extremes are 3 and 45, and the common difference 2, what is the sum of the series? Ans. 528. 13. A owes B a certain sum, to be discharged in a year, by paying 6 cents the first week, and 18 cents the second week, and thus to increase every week by 12 cents till the last payment should be $6.18. What is the debt? Ans. $162.24. PROBLEM V. The extremes and sum of the series given, to find the common difference. RULE. Divide the product of the sum and the difference of the extremes, by the difference of twice the sum of the series, and the sum of the extremes, and the quotient will be the common differ ence. 14. The extremes are 3 and 45, and the sum of the series 528. What is the common difference? Ans. 2. |