Explanation: If we divide the total area by 4, we get 10,000 as the area of a square having the breadth b on each side. From this we find the breadth or width b to be the square root of 10,000 or 100 ft. If the length is four times as great it will be 400 ft. and the dimensions of the building will be 400 by 100. 83. Cube Root.-The Cube Root of a given number is another number which, when cubed, produces the given number. In other words, the cube root is one of the three equal factors of a number. The cube root of 8 is 2, because 23=2×2×2=8; also the cube root of 27 is 3 (since 33 = 27) and the cube root of 64 is 4 (since 43 = 64). The sign of cube root is placed over the number of which we want the root. Thus we would write 28=2 √64=4 1000=10 If we consider the number of which we want the cube root as representing the volume of a cubical block, then the cube root of the number will represent the length of one edge of the cube. The cube root of 1728 is 12 and a cube containing 1728 cu. in. will measure 12 in. on each edge. There are four ways of getting cube roots: (1) by actual calculation, (2) by reference to a table of cubes or cube roots, (3) by the use of logarithms, and (4) by the use of some calculating device like the slide rule. The use of a table is the simplest way of finding cube roots, but its value and accuracy is limited by the size of the table. Tables of cubes or cube roots are to be found in many handbooks and catalogues and should be used whenever they give the desired root with sufficient accuracy. Logarithms give us an easy way of getting cube roots, but here also a table is necessary and the accuracy is limited by the size of the table of logarithms. The use of logarithms will be explained in a later chapter. The ordinary pocket slide rule will give the first three figures of a cube root and for many calculations this is sufficiently accurate. The method of actually calculating cube roots is very complicated and is used so seldom that one can never remember it when he needs it. Consequently, if it is necessary to hunt up a book to find how to extract the cube root, one might just as well look up a table of cube roots or a logarithm table, either of which will give the root much quicker. The next chapter contains tables of cube roots and a chapter further on explains the use of logarithms. Note. These answers are given so that the student can see if he understands the operations of square root before proceeding further. 142. The two sides of a right triangle are 36 and 48 ft.; what is the length of the hypotenuse? 143. A square nut for a 2 in. bolt is 3 in. on each side. What is the length of the diagonal, or distance across the corners? 144. A steel stack 75 ft. high is to be supported by 4 guy wires fastened to a ring two-thirds of the way up the stack and having the other ends anchored at a distance of 50 ft. from the base and on a level with the base. How many feet of wire are necessary, allowing 20 ft. extra for fastening the ends? 145. What would be the diameter of a circular brass plate having an area of 100 sq. in.? -17-6 20 FIG. 33. 146. A lineshaft and the motor which drives it are located in separate rooms as shown in Fig. 33. Calculate the exact distance between the centers of the two shafts. 147. I want to cut a rectangular sheet of drawing paper to have an area of 235 sq. in. and to be one and one-half times as long as it is wide. What would be the dimensions of the sheet? 148. A 6 in. pipe and an 8 in. pipe both discharge into a single header. Find the diameter of the header so that it will have an area equal to that of both the pipes. 149. What would be the diameter of a 1 lb. circular cast iron weight in. thick? 150. How long must be the boom in Fig. 34 to land the load on the 12 ft. pedestal, allowing 4 ft. clearance at the end for ropes, pulleys, etc.? 27FIG. 34. 21 CHAPTER XII MATHEMATICAL TABLES (CIRCLES, POWERS, AND ROOTS) 84. The Value of Tables.-There are certain calculations that are made thousands of times a day by different people in different parts of the world. For example, the circumferences of circles of different diameters are being calculated every day by hundreds and thousands of men. To save much of the time that is thus wasted in useless repetition, many of the common operations and their results have been "tabulated," that is, arranged in tables in the same way as are our multiplication tables in arithmetics. These tables are not learned, however, as were the multiplication tables, but are consulted each time that we have need for their assistance. Just what tables one needs most, depends on his occupation. The machinist has use for tables of the decimal equivalents of common fractions, tables of cutting speeds, tables of change gears to use for screw cutting, etc. The draftsman would use tables of strengths and weights of different materials, safe loads for bolts, beams, etc., tables of proportions of standard machine parts of different sizes, etc. The engineer uses tables of the properties of steam, and of the horse-power of engines, boilers, etc. There are certain mathematical tables that are of value to nearly everyone. Among these are the tables given in this chapter: Tables of Circumferences and Areas of Circles; Tables of Squares, Cubes, Square Roots, and Cube Roots of Numbers. 85. Explanation of the Tables.-The first table is to save the necessity of always multiplying the diameter by 3.1416 when we want the circumference of a circle, or of squaring the diameter and multiplying by .7854 when the area of a circle is wanted. To find the circumference of a circle: Find, in the diameter column, the number which is the given diameter; directly across, in the next column to the right, will be found the corresponding circumference. Examples: Diameter 11 Circumference 3.9270 Diameter 90, To find the area of a circle: Find, in the diameter column, the number which is the given diameter; directly across, in the second column to the right (the column headed "Area") will be found the area. Examples: Diameter 66, Area 3421.2 If the area or circumference is known and we want to get the diameter, we find the given number in the area or circumference column and read the diameter in the corresponding diameter column to the left. The second table, that of squares, cubes, square roots, and cube roots, is especially valuable in avoiding the tedious process of extracting square or cube roots. The table is read the same as the other one. Find the given number in the first column; on a level with it, in the other columns, will be found the corresponding powers and roots, as indicated in the headings at the tops of the columns. 86. Interpolation. This is a name given to the process of finding values between those given in the tables. For example, suppose we want the circumference of a 301 in. circle. The table gives 30 and 301⁄2 and, since 30 is half way between these, its circumference will be half way between that of a 30 in. and a 30 in. circle. Then the circumference of the 301 in. circle is just half this difference more than that of the 30-in. circle, This method enables us to increase greatly the value of tables. For most purposes the interpolation can be done quickly, and while it requires some calculating, is much shorter than the complete calculation would be. This is especially true in finding square or cube roots. Example: Find from the table the cube root of 736.4 3/737=9.0328 3/736=9.0287 Difference 41 .4X the difference = .4X41=16.4 9.0287 /736.49.0303, Answer. Explanation: The root of 736.4 will be between that of 736 and that of 737, and will be .4 of the difference greater than that of 736. In making this correction for the .4, we forget, for the minute, that the difference is a decimal and write it as 41 merely to save time. We then multiply it by .4, and, dropping the decimal part, add the 16 to the 90287. This gives 9.0303 as the cube root of 736.4 Hence, 736.4-9.0303 |