Applications. 1. What must be the dimensions of a cubical bin, that its volume or capacity may be 19683 feet ? 2. If a cubical body contains 6859 cubic feet, what is the length of one side? what the area of its surface? 3 The volume of a globe is 46656 cubic inches; what would be the side of a cube of equal solidity? 4. A person wishes to make a cubical cistern, which shail hold 150 barrels of water: what must be its depth? 5. A farmer constructed a bin that would contain 1500 bushels of grain; its length and breadth were equal, and each half the height: what were its dimensions? 6 What is the difference between half a cubic yard, and a cube whose edge is half a yard? 7. A merchant paid $911.25 for some pieces of muslin. He paid as many cents a yard as there were yards in each piece, and there were as many pieces as there were yards in one piece: how many yards were there, and how much did he pay a yard? 8. If a sphere 3 feet in diameter contains 14.1372 cubic feet, what are the contents of a sphere 6 feet in diameter? 33 : 63 :: 14.1372 : 113.0976. Ans. 9. If a ball 2 inches in diameter weighs 8 pounds, bow much will one of the same kind weigh, that is 5 inches in diameter? 10. What must be the size of a cubical bin, that will conthin 8 times as much as one that is 4 feet on a side? 11. How many globes, 6 inches in diameter, would be required to make one 12 inches in diameter? 12 If a ball of silver, one unit in diameter, is worth $8, what will be the value of one 5 units in diameter ? 13. If a plate of silver, 6 inches long, 3 inches wide and inch thick, is worth $100, what will be the dimensions of a similar plate, of the same metal, worth $800? 14. If a man can dig a cellar 12 feet long, 10 feet wide, and 4 feet deep, in 3 days, what will be the dimensions of a similar cellar, requiring 24 days to dig it, working at the same rate, and the ground being of the same degree of hardness? 15 If I put 2 tons of hay in a stack 10 feet high, how high must a similar stack be to contain 16 tons? 16. Four women bought a ball of yarn 6 inches in diameter, and agreed that each should take her share separately from the outer part of the ball: how much of the diameter did each wind off? ARITHMETICAL PROGRESSION. 392. An ARITHMETICAL PROGRESSION is a series of numbers in which each is derived from the one preceding, by the addition or subtraction of the same number. THE COMMON DIFFERENCE is the number which is added or subtracted. 393. When the series is formed by the continued addition of the common difference, it is called an increasing series; and when it is formed by the subtraction of the common difference, it is called a decreasing series: thus, 2, 5, 8, 11, 14, 17, 20, 23, 23, 20, 17, 14, 11, 8, 5, 2, is an increasing series; is a decreasing series. The several numbers are called terms of the progression. The first and last terms are called the extremes, and the in termediate terms are called the means. 394. In every arithmetical progression there are five parts, any three of which being given or known, the remaining two can be determined. They are, 1st, The first term; 2d, The last term; 3d, The common difference; 4th, The number of terms; 5th, The sum of all the terms. CASE I. 395. Having given the first term, the common difference, and the number of terms, to find the last term. 1. The first term of an increasing progression is 4, the common difference 3, and the number of terms 10: what is the last term? 9 OPERATION. No. less 1 3 com. diff. 27 4 1st term. last term. 31 ANALYSIS. By considering the manner in which the increasing progression is formed, we see that the 2d term is obtained by adding the common difference to the 1st term; the 3d, by adding the common difference to the 2d; the 4th, by adding the common difference to the 3d, and so on; the number of additions, in every case, being one less than the number of terms found. Instead of making the additions, we may multiply the common difference by the number of additions, that is, by 1 less than the number of terms, and add the first term to the product. Rule.-Multiply the common difference by 1 less than the number of terms: if the progression is increasing, add the product to the first term, and the sum will be the last term; if it is decreasing, subtract the product from the first term, and the difference will be the last term. Examples. 1. What is the 18th term of an arithmetical progression, of which the first term is 4, and the common difference 5? 2. A man is to receive a certain sum of money in 12 pay. ments the first payment is $300, and each succeeding pay ment is less than the previous one by $20: what will be the last payment? 3. What will $200 amount to in 15 years, at simple interest, the increase being $14 for the first year, $28 for the econd, and so on? 4. Mr. Jones has 12 children. He gives, by will, $1000 to the youngest, $50 more to the next older, and so on to each next older $50: how much did the oldest receive? 5. A man has a piece of land 35 rods in length, which tapers to a point, and is found to increase rod in width, for every rod in length: what is the width of the wide end? 6. James and John have 100 marbles. It is agreed between them that John shall have them all, if he will place them in a straight line half a foot apart, and so that he shall be obliged to travel 300 feet to get and bring back the furthest marble; and also, if he will tell, without measuring, how far he must travel to bring back the nearest. How far? CASE II. 396. Knowing the two extremes of an arithmetical progression, and the number of terms, to find the common difference. 1. The two extremes of a progression are 4 and 68, and the number of terms 17: what is the common difference? OPERATION. 68 4 ANALYSIS.-Since the common difference multiplied by 1 less than the number of terms gives a product equal to the difference of the extremes, if we divide the difference of the extremes by 1 less than the number of terms, the quotient will be the common difference : Lence, 17-116) 64(1 Rule.-Subtract the less extreme from the greater, and divide the remainder by 1 less than the number of terms: the quotient will be the common difference. Examples. 1. A man started from Chicago and traveled 15 days; each day's journey was longer than that of the preceding day by the distance which he traveled the first day: what was hi daily increase if he traveled 75 miles the last day? A merchant sold 14 yards of cloth, in pieces of 1 yard each; for the first yard he received $, and for the last $264: what was the difference in the price per yard? 3. A board is 17 feet long; it is 2 inches wide at one end, and 14 at the other: what is the average increase in width per foot in length? 4. The fourth term of a series is 12, and the eleventh is 33 find the intermediate terms. CASE III. 397. To find the sum of the terms of an arithmetical progres sion. 1. What is the sum of the series whose first term is 2, common difference 3, and the number of terms 8? Given series, 20 23 2 5 8 11 14 17 20 17 14 11 8 5 2 25+25+25 + 25 + 25 + 25 + 25 +25 Same, order inverted, 23 ANALYSIS.-The two series are the same; hence, their sum is equal to twice the given series But their sum is equal to the sum of the two extremes, 2 and 23, taken as many times as there are terms; and the given series is equal to half this sum, or to t sum of the extremes multiplied by half the number of terins. Rule. Add the extremes together, and multiply their sum by half the number of terms; the product will be the sum of all the terms. |