on one side 33 feet, and on the other side 21 feet, from the ground. What is the breadth of the street? Ans. 56.64+ feet. 10. Two men start from one corner of a park one mile square, and travel at the same rate. A goes by the walk around the park, and B takes the diagonal path to the opposite corner, and turns to meet A at the side. How many rods from the corner will the meeting take place? Ans. 93.7+ rods. 11. The top of a castle is 45 yards high, and the castle is surrounded by a ditch 60 yards wide. What would be the length of a rope that would reach from the outside of the ditch to the top of the castle? Ans. 75 yards. 12. A ladder 52 feet long stands close against the side of a building. How many feet must it be drawn out at the bottom, that the top may be lowered 4 feet? 13. A room is 20 feet long, 16 feet wide, and 12 feet high. What is the distance from one of the lower corners to the opposite upper corner? Ans. 28.284271+ feet. 14. It requires 63.39 rods of fence to inclose a circular field of 2 acres. What length will be required to inclose 3 acres in circular form? Ans. 77.63+ rods. 15. The radius of a certain circle is 5 feet. What will be the radius of another circle containing twice the area of the first? Ans. 7.07106+ feet. 16. A certain circular race-track has a diameter of 1500 feet. What would be the diameter of a similar track 4 times as large ? Ans. 3000 feet. 17. If it costs $167.70 to inclose a circular pond containing 17 A. 110 P., what will it cost to inclose another as large? Ans. $75. 18. If a cistern 6 ft. in diameter holds 80 bbl. of water, what is the diameter of a cistern of the same depth that holds 1200 bbl. ? CUBE ROOT. 439. The Cube Root of a number is one of the three equal factors that produce the number. Thus, the cube root of 27 is 3, since 3 × 3 × 3 = 27. In extracting the cube root, the first thing to be determined is the relative number of places in a cube and its root. The law governing this relation is exhibited in the following examples: From these examples, we perceive, 1. That a root consisting of 1 place may have from 1 to 3 places in the cube. 2. That in all cases the addition of 1 place to the root adds three places to the cube. If we point off a number into three-figure periods, commencing at the right hand, the number of full periods and the left hand full or partial period will indicate the number of places in the cube root, the highest period corresponding to the highest figure of the root. To ascertain the relations of the several figures of the root to the periods of the number, decompose any number, as 5423. The cube of the first figure of the root is contained wholly in the first period of the power: the cube of the first two figures in the first two periods of the power, etc. EXAMPLES. 40. 1. What is the length of one side of a cubical block containing 413494 solid inches? OPERATION COMMENCED. 41349474 343 14700 70494 SOLUTION. Since the block is a cube, its side will be the cube root of its solid contents, which we will proceed to compute. Pointing off the given number, the two periods show that there will be two figures, tens and units, in the root. The tens of the root must be extracted from the first period, 413 thousands. The greatest cube in 413 thousands is 343 thousands, the cube of 7 tens; we therefore write 7 tens in the root at the right of the given number. Since the entire root is to be the side of a cube, let us form a cubical block (Fig. I), the side of which is 70 inches in length. The contents of this cube are 70 x 70 x 70 = 343000 solid inches, which we subtract from the given number. This is done in the operation by subtracting the cube number, 343, from the first period, 413, and to the remainder bringing down the second period, making the entire remainder 70494. FIG. I. If we now enlarge our cubical block (Fig. I) by the addition of 70494 solid inches, in such a manner as to preserve the cubical form, its size will be that of the required block. To preserve the cubical form, the addition must be made upon three adjacent sides or faces. The addition will therefore be composed of 3 flat blocks to cover the 3 faces (Fig. II); 3 oblong blocks to fill the vacancies at the edges (Fig. III); and 1 small cubical block to fill the vacancy at the corner (Fig. IV). Now, the thickness of this enlargement will be the additional length of the side of the cube, and, consequently, the second figure in the root. To find thickness, we may divide solid contents by surface, or area. But the area of the 3 oblong blocks and little cube cannot be found till the thickness of the addition is determined, because their common breadth is equal to this thickness. We will, therefore, find the area of the 3 flat blocks, which is sufficiently near the whole area to be FIG. II. used as a trial divisor. As these are each equal in length and breadth to the side of the cube whose faces they cover, the whole area of the three is 70 x 70 x 3 = 14700 square inches. This number is obtained in the operation by annexing 2 ciphers to three times the square of 7; the result being written at the left hand of the dividend. Dividing, we obtain 4, the probable thickness of the addition, and second figure of the root. With this assumed figure, we will complete our divisor by adding the area of the 4 blocks, before undetereach 70 inches long; and the its dimensions to the thickness mined. The 3 oblong blocks are little cube, being equal in each of of the addition, must be 4 inches long. Hence, their united length is 70+ 70+70+ 4 = 214. This number is obtained in the operation by multiplying the 7 by 3, and annexing the 4 to the product, the result being written in column I, on the next line below the trial divisor. Multiplying 214, the length, by 4, the common width, we obtain 856, the area of the four blocks, which added to 14700, the trial divisor, makes 15556, the complete divisor; and multiplying this by 4, the second figure in the root, and subtracting the product from the dividend, we obtain a remainder of 8270 solid inches. With this remainder, for the same reason as before, we must proceed to make a new enlargement. But since we have already two figures FIG. III. in the root, corresponding to the two periods of the given number, the next figure of the root must be a decimal; and we therefore annex to the remainder a period of three decimal ciphers, making 8270.000 for a new dividend. OPERATION I. II. CONTINUED. 413494 | 74 343 14700 70494 62224 8270.000 214 856 15556 The trial divisor to obtain the thickness of this second enlargement, or the next figure of the root, will be the area of three new flat blocks to cover the three sides of the cube already formed; and this surface (Fig. IV) is composed of 1 face of each of the flat blocks already used, 2 faces of each of the oblong blocks, and 3 faces of the little cube. But we have in the complete divisor, 15556, 1 face of each of the flat blocks, oblong blocks, and little cube; and in the correction of the trial divisor, 856, 1 face of each of the oblong blocks and of the little cube; and in the square of the last root figure, 16, a third face of the little cube. Hence, 16+ 856+15556 = 16428, the significant figures of the new trial divisor. This number is obtained in the operation by adding the square of the last root figure mentally, and combining units of like OPERATION-CONTINUED. FIG. IV. order, thus: 16, 6, and 6 are 28, and we write the unit figure in the new trial divisor; then 2 to carry, and 5 and 5 are 12, etc. |