Using 2d trial units' figure, 5: 3(202 × 5) = 6000 3(20 × 52) = 1500 58= 125 Trial divisor, or 3 x t2 1200 3 x t x u 300 u2 257625 Complete divisor 1525 This is called a trial figure because as we have disregarded the other two parts of the cube, and considered 7625 as made up of only one of the three parts, our trial divisor may be found to be too large when we come to consider the two parts we disregarded for the purpose of finding the units' figures. Regarding 7625, then, as made up of 3 times the square of the tens multiplied by the units, we see that two of these factors are known, namely: 3, and the tens which we have found to be 2, or 20 units. The square of the tens is 20 × 20, or 400. 3 times the square of the tens is 1200. The units' figure is not known, but may be found approximately by dividing 7625 by 1200 as a trial divisor, or, what is the same thing, ignoring the zeros and dividing 76 by 12. The quotient is 6, our trial units' figure. Before writing the 6 in the root, we should test to see whether it is the proper figure. Writing out the value of each of the three parts as shown at B, p. 323, we find their sum is 9576, or more than 7625. We therefore conclude that our trial units' figure, 6, is too large. Substituting for it the next smaller number, 5, and writing out the value of the three parts as before, we find their sum is 7625, showing that 5 is the correct units' figure, which we place in the root over the second period. In the process we have employed in extracting the cube root of numbers, we have shortened the above operation as follows: After dividing 7625 by 1200 as a trial divisor, and finding the trial units' figure, 6, we may complete the divisor by adding to it 3(20 × 6), òr 360, and 62, or 36, making 1200 + 360 + 36, or 1596, which equals 3 × ť2 + 3(t × u) + u2. Each of these three parts lacks the same factor, the units. On multiplying their sum, 1596, called the complete divisor, by 6, the product, 9576, is the same as we found the sum of the three full parts to be, in testing to determine whether 6 was the correct units' figure. In testing 5 for the units' figure, we proceed in the same way: Adding to 1200, 3(20 × 5), or 300, and 52, or 25, we have 1525, the complete divisor, which, when multiplied by the units' figure, 5, gives 7625, the sum of the three parts of the cube remaining after the cube of the tens, 8000, has been subtracted. The cube subtracted from the first period, and the product of the complete divisor by the second figure of the root, together equal the entire number or cube. If there were three figures in the root, these two numbers together would equal the cube of the first two figures of the root considered as tens, and the units' figure would be found in the same way as the tens' figure was found, and so on, if the root had more than three figures. 2. Extract the cube root of 97,336, explaining the process. Solve the example, answering each question in order. The information necessary for answering these questions may be obtained by a careful study of the explanation of the process of extracting the cube root, pp. 321-325. 1. Why do you point off the number into periods of three figures each, beginning at units' place? 2. How is the first figure of the root found? Why? What is it? 3. The number subtracted from the first period is what as related to the first figure of the root? 4. The number formed by annexing the next period to the remainder contains what three parts of the entire cube? 5. What part of the entire cube is it regarded as being for the purpose of finding the second figure of the root? 6. What factors does this part contain? 7. Which of these factors are known? 8. When the product of these factors and two of the factors are known, how can you find the other factor? 9. What is done in this case to find the trial second figure of the root? What is this figure? 10. What is the trial divisor? Of what factors is it the product? What two additions are made to it to form the complete divisor? Give the numbers added, and state in general terms of what factors each number is the product. 11. State the terms of which the complete divisor is composed. 12. The product of the complete divisor by the trial units' figure of the root is composed of what terms of the cube? 13. What is done with this product? What does this operation show ? 3. Extract the cube root of 148,877, explaining each step in the operation. COMMON MEASURES TROY WEIGHT 215. Troy weight is used in weighing gold, silver, and jewels. 24 grains (gr.) = 1 pennyweight (pwt.) NOTES. In indicating the fineness or purity of gold, it is considered as composed of 24 parts, called carats, and the number of carats specified is the number of 24ths of pure gold that it contains. The difference between this number and 24 is the number of parts of alloy, or less valuable metal, combined with the gold. An eighteen-carat ring is pure The weight of diamonds is usually given in carats, each carat weighing 3 grains (nearly). A six-carat diamond weighs 20 grains Troy, nearly. APOTHECARIES' WEIGHT 216. Apothecaries' weight is used by physicians in prescribing, and by apothecaries in compounding medicines. = 1 oz. Apothecaries' weight 480 grains Drugs are sometimes bought by druggists in bulk by Avoirdupois weight, but are sold by Apothecaries' weight, as smaller subdivisions than ounces and drams are frequently needed in compounding medicines to fill physicians' prescriptions. The pound Avoirdupois is heavier than either the Troy or Apothecaries' pound. The ounce Avoirdupois is lighter than either the Troy or Apothecaries' ounce. What unit is common to the three weights? Avoirdupois pounds may be reduced to grains by multiplying 7000, the number of grains in 1 lb., by the number of pounds. Avoirdupois ounces may be reduced to grains by multiplying 437, the number of grains in 1 oz., by the number of ounces. Any number of grains may be reduced to higher denominations of Apothecaries' or Troy weight by the process of Reduction Ascending. Any number of grains may be reduced to pounds and fractions of a pound, Avoirdupois, by dividing the number of grains by 7000; to ounces, Avoirdupois, by dividing the number of grains by 437.5. WRITTEN 1. How many grains heavier is a pound Avoirdupois than a pound Troy? 2. Which is the heavier, a pound of feathers or a pound of gold, and how much heavier ? 3. Express 2 lb. Avoirdupois in the denominations of Apothecaries' weight. 4. Express 3 lb. 6 oz. Avoirdupois in the denominations of Apothecaries' weight. 5. How many ounces are there in 32 3? In 25 3? 6. How many drams are there in 12 ? In 15 ? 7. How many grains are there in 5 3 and 3 3? 8. Reduce 2465 gr. to higher denominations, Apothecaries' weight. 9. What part of a pound Avoirdupois is a pound Troy? 10. What part of an ounce Troy is an ounce Avoirdupois ? 11. Reduce 2465 gr. to higher denominations, Troy weight. United States gold and silver coins are .9 pure gold or silver and .1 copper. The weight of a silver dollar is 412 grains. The weight of the 10-dollar gold piece is 258 grains. 12. What is the weight of the pure silver in 100 silver dollars? 13. What is the weight of the pure gold in a 10-dollar gold piece? 14. If 100 pounds of pure gold are to be coined into eagles, how much copper is needed as alloy? 15. When the necessary amount of copper has been added to 100 pounds of pure gold, what is the value of the metal in dollars? SURVEYORS' LINEAR MEASURE 217. This measure is used by surveyors in measuring distances. 7.92 inches (in.) 100 links, or 4 rods = 1 link (1.) The chain referred to in the foregoing is called Gunter's chain; its length is 66 ft., or 792 inches. Its divisions are decimal, so that chains and links may be written as chains and decimals of chains; thus, 15 links equals .15 of a chain; 25 chains 15 links is written 25.15 chains. ORAL 1. What part of a chain is 1 link? 5 links? 18 links? 2. What part of a chain are 10 links? 1 rod? 3. How many chains are there in mile? 4. How many feet are there in 25 links? What part of In mile? & In 50 links? |