11. There is a circle, whose diameter is 4 inches; what is the diameter of a circle 9 times as large? Note. The areas or contents of circles are in proportion to the squares of their diameters, or of their circumferences. Therefore, to find the diameter required, square the given diameter, multiply the square by the given ratio, and the square root of the product will be the diameter required. 4X4 X 9= 12 inches, Ans. 12. There are two circular ponds in a gentleman's pleasure ground; the diameter of the less is 100 feet, and the greater is 3 times as large; what is its diameter? Ans. 173'2+ feet. 13. If the diameter of a circle be 12 inches, what is the diameter of one as large ? Ans. 6 inches. ¶ 109. 14. A carpenter has a large wooden squarc; one part of it is 4 feet long, and the other part 3 feet long; what is the length of a pole, which will just reach from one end to the other? Note. A figure of 3 sides is called a triangle, and, if one of the corners be a square corner, or right angle, like the angle at B in the annexed figure, it is called a right-angled triangle, of which the square of the longest side, A C, (called the hypotenuse,) is equal to the sum of the squares of the other two sides, AB and B C. 42 = 16, and 32 9; then, 9+16 5 feet, Ans. 15. If, from the corner of a square room, 6 feet be measured off one way, and 8 feet the other way, along the sides of the room, what will be the length of a pole reaching from point to point? Ans. 10 feet. 16. A wall is 32 feet high, and a ditch before it is 24 feet wide; what is the length of a ladder that will reach from the top of the wall to the opposite side of the ditch? Ans. 40 feet. 17. If the ladder be 40 feet, and the wall 32 feet, what is the width of the ditch? Ans. 24 feet. Ans. 32 feet 19. The distance between the lower ends of two equal rafters is 32 feet, and the height of the ridge, above the beam on which they stand, is 12 feet; required the length of each rafter. Ans. 20 feet. 20. There is a building 30 feet in length and 22 feet in width, and the eaves project beyond the wall 1 foot on every side; the roof terminates in a point at the centre of the building, and is there supported by a post, the top of which is 10 feet above the beams on which the rafters rest; what is the distance from the foot of the post to the corners of the eaves? and what is the length of a rafter reaching to the middle of one side? a rafter reaching to the middle of one end? and a rafter reaching to the corners of the eaves? Answers, in order, 20 ft.; 15'62 + ft.; 18'86 + ft.; and 22'36+ ft. 21. There is a field 800 rods long and 600 rods wide; what is the distance between two opposite corners ? Ans. 1000 rods. 22. There is a square field containing 90 acres; how many rods in length is each side of the field? and how rods apart are the opposite corners ? many Answers, 120 rods; and 1697 rods. 23. There is a square field containing 10 acres; what distance is the centre from each corner? Ans. 28'28+ rods, EXTRACTION OF THE CUBE T110. A solid body, having six equal sides, and each of the sides an exact square, is a CUBE, and the measure in length of one of its sides is the root of that cube; for the length, breadth and thickness of such a body are all alike; consequently, the length of one side, raised to the 3d power, gives the solid contents. (See ¶ 36.) Hence it follows, that extracting the cube root of any number of feet is finding the length of one side of a cubic body, of which the whole contents will be equal to the given number of feet. 1. What are the solid contents of a cubic block, of which each side measures 2 feet? Ans. 232 X 2 X 28 feet. 2. How many solid feet in a cubic block, measuring 5 feet on each side? Ans. 53 125 feet. 3. How many feet in length is each side of a cubic block, containing 125 solid feet? Note. The root may be found by trial. feet? feet? 27 solid feet? Ans. 125 5 feet. 512 solid 4. What is the side of a cubic block, containing 64 solid 216 solid feet?Answers, 4 ft.; 3 ft.; 6 ft.; and 8 ft. 5. Supposing a man has 13824 feet of timber, in separate blocks of 1 cubic foot each; he wishes to pile them up in a cubic pile; what will be the length of each side of such a pile? It is evident, the answer is found by extracting the cube root of 13824; but this number is so large, that we cannot so easily find the root by trial as in the former examples ;We will endeavour, however, to do it by a sort of trial; and, 1st. We will try to ascertain the number of figures, of which the root will consist. This we may do by pointing the number off into periods of three figures each (T 107, ex. 1.) OPERATION. 13824 (2 8 5824 FIG. I. 20 Pointing off, we see, the root will consist of two figures, a ten and a unit. Let us, then, seek for the first figure, or tens of the root, which must bé extracted from the left hand period, 13, (thousands.) The greatest cube in 13 (thousands) we find by trial, or by the table of powers, to be 8, (thousands,) the root of which is 2, (tens;) therefore, we place 2 (tens) in the root. The root, it will be recollected, is one side of a cube. Let us, then, form a cube, (Fig. I.) each side of which shall be supposed 20 feet, expressed by the root now obtained. The contents of this cube are 8000 feet, Contents. 20X20X20 8000 solid feet, which are now disposed of, and which, consequently, are to be deducted from the whole number of feet, 13824. 8000 taken from 13824 leave 5824 feet. This deduction is most readily performed by subtracting the cubic number, 8, or the cube of 2, (the figure of the root already found,) from 20 the period 13, (thousands,) and bringing down the next period by the side of the remainder, making 5824, as before. = 2d. The cubic pile A D is now to be enlarged by the addition of 5824 solid feet, and, in order to preserve the cubic form of the pile, the addition must be made on one half of its sides, that is, on 3 sides, a, b, and c. Now, if the 5824 solid feet be divided by the square contents of these 3 equal sides, that is, by 3 times, (20 X 20400) 1200, the quotient will be the thickness of the addition made to each of the sides a, b, c. But the root, 2, (tens,) already found, is the length of one of these sides; we therefore square the root, 2, (tens,) = 20 X 20 400, for the square contents of one side, and multiply the product by 3, the number of sides, 400 X 3 1200; or, which is the same in effect, and more convenient in practice, we may square the 2, (tens,) and multiply the product by 300, thus, 2 x2=4, and 4 X 300-1200, for the divisor, as before. = = The divisor, 1200, is con OPERATION-CONTINUED. tained in the dividend 4 times; consequently, 4 feet is the = 16, by the length of all the deficiencies, that is, by 3 times T == the length of each side, which is expressed by the former quotient figure, 2, (tens.) 3 times 2 (tens) are 6 (tens) 60; or, what is the same in effect, and more convenient in practice, we may multiply the quotient figure, 2, (tens,) by 30, thus, 2 X 30=60, as before; then, 60 X 16=960, contents of the three deficiencies n, n, n. Looking at Fig. III., we perceive there is still a deficiency in the corner where the last blocks meet. This deficiency is a cube, each side of which is equal to the last quotient figure, 4. The cube of 4, therefore, (4 X 4 X 464,) will be the solid contents of this corner, which in Fig. IV. is seen filled. Now, the sum of these several additions, viz. 4800 + 960645824, will make the subtrahend, which, subtracted from the dividend, leaves no remainder, and the work is done. Fig. IV. shows the pile which 13824 solid blocks of one foot each would make, when laid together, and the root, 24, shows the length of one side of the pile. The correctness of the work may be ascertained by cubing the side now found, 243, thus, 24 X 24 X 24 = 13824, the proved by adding together given number; or it may be Feet, 8000 contents of Fig. I. 4800 960 64 addition to the sides a, b, and c, Fig. 1. addition to fill the deficiencies n, n, n, Fig. II. addition to fill the corner e, e, e, Fig. IV. 13824= contents of the whole pile, Fig. IV., 24 feet on ach side |