RULE. Extract the square root of half the square of the diameter. Or, Multiply the diameter by .7071. 3. The circumference of a circle is 104 yards. Find the side of the inscribed square. Ans. 23.4 yd. +. 755. To find the area of a circular ring, formed by two concentric circles. 1. Find the area of a circular ring, when the diameters of the circles are 20 and 30 feet. OPERATION.—(30 + 20 x 30 – 20) x .7854 = 392.7 sq. ft., area. 2. Find the area of a circular ring formed by two concentric circles, whose diame rs are 7 ft. 9 in, and 4 ft. 3 in. Ans. 32.9868 sq. ft. RULE.—Multiply the sum of the two diameters by their difference, and the product by .7854; the result is the area. 3. Two diameters are 35.75 and 16.25 feet; find the area of the Ans. 796.39 sq. ft. 4. The area of a circle is 1 A. 154.16 P. In the center is a pond of water 10 rods in diameter; find the area of the land and of the water. Ans. Land, 235.62 P.; water, 78.54 P. ring 756. To find a mean proportional between two numbers. 1. What is a mean proportional between 3 and 12 ? OPERATION.-V12 3 = 6, the mean proportional. When three numbers are proportional, the product of the extremes is equal to the square of the mean. RULE. Extract the square root of the product of the two numbers. Find a mean proportional between 2. 36 and 81. 4. 64 and 12.25. 6. and 1 3. 42 and 168. 5. 8 and 288. 7. 4 and 144 8. A tub of butter weighed 36 lb. by the grocer's scales; but being placed in the other scale of the balance, it weighed only 30 lb. What was the true weight of the butter ? Ans. 32.86 + lb. SIMILAR PLANE FIGURES. 757. Similar Plane Figures are such as have the same form; or have angles equal each to each, the same number of sides, and the sides containing the equal angles proportional. All circles, squares, equiangular triangles, and regular polygons of the same number of sides are similar figures. The like dimensions of circles, that is, their radii, diameters, and circumferences, are proportional. PRINCIPLES. 1. The like dimensions of similar plane figures are proportional. 2. The areas of similar plane figures are to each other as the squares of their like dimensions. And conversely, 3. The like dimensions of similar plane figures are to each other as the square root of their areas. The same principles apply also to the surfaces of all similar figures, such as triangles, rectangles, etc. ; the surfaces of similar solids, as cubes, pyramids, etc.; and to similar curved surfaces, as of cylinders, cones, and spheres. Hence, 4. The surfaces of all similar figures are to each other as the squares of their like dimensions. And conversely, 5. Their dimensions are as the square roots of their surfaces. PROBLEMS. 1. A triangular field whose base is 12 ch. contains 2 A. 80 P. Find the area of a field of similar form whose base is 48 ch. OPERATION.—122 : 482 :: 2 A. 80 P.: x= 6400 P= 40 A., area. (PRIN. 2.) 2. The side of a square field containing 18 A. is 60 rd. long. Find the side of a similar field that contains $ as many acres. OPERATION.–18 A. : 6 A. :: 602 : 22 =1200 ; 1200 : = 34.64 rd. +, side. (PRIN. 3.) 3. Two circles are to each other as 9.to 16; the diameter of the less being 112 feet, what is the diameter of the greater? OPERATION.-9:16 :: 1122 : x=3:4:: 112 : x=149 ft. 4 in., diameter. (PRIN. 2.) 4. A peach orchard contains 720 sq. rd., and its length is to its breadth as 5 to 4; what are its dimensions ? OPERATION.—The area of a rectangle 5 by 4 equals 20 (745). 20 : 720 :: 52 : x2 = 900; 7900 = 30 rd., length. 5. It is required to lay out 283 A. 107 P. of land in the form of a rectangle, so that the length shall be 3 times the width. Find the dimensions. Ans. 369 rd. ; 123 rd. 6. A pipe 1.5 in. in diameter fills a cistern in 5 hr.; find the diameter of a pipe that will fill the same cistern in 55 min. 6 sec. Ans, 3.5 in. 7. The area of a triangle is 24276 sq. ft., and its sides in proportion to the numbers 13, 14, and 15. Find the length of its sides in fcet. Ans. 221, 238, and 255 ft. 8. A field containing 6 A. is laid down on a plan to a scale of 1 in. to 20 ft. How much paper will it cover ? Ans. 653.4 sq. in. 9. If it cost $167.70 to enclose a circular pond containing 17 A. 110 P., what will it cost to enclose another | as large? Ans. $75. 10. 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. 11. If 63.39 rd. of fence will enclose a circular field containing 2 A., what length will enclose 8 A. in circular form? Ans. 126.78 rd. 758. REVIEW OF PLANE FIGURES. PROBLEMS. 1. The area of a triangle is 270 yd., and the perpendicular 45 ft. Find the base. 2. Find the area of a square whose perimeter is the same as that of a rectangle 48 ft. by 28 feet. 3. A rectangle whose length is 3 times its width contains 1323 P. Find its dimensions. Ans. 21 rd. by 63 rd. 4. Find the area of an equilateral triangle whose sides are 36 ft. 5. The area of a circle is 7569 square feet. Find the length of the side of a square of equal area. Ans. 87 ft. 6. How much less will the fencing of 20 A. cost in the square form than in the form of a rectangle whose breadth is the length, the price being $2.40 per rod ? Ans. $185.43. 7. A house that is 50 ft. long and 40 ft. wide has a square or pyramidal roof, whose height is 15 ft. Find the length of a rafter reaching from a corner of the building to the vertex of the roof. 8. Find the length of a rafter reaching from the middle of one side. Ans. 25 ft. 9. Find the length of a rafter reaching from the middle of one end, 10. What is the diameter of a circular island containing 14 sq. miles ? Ans. 403.7 rd. 11. How many rods more of fencing are required to enclose a square field whose area is 5 acres, than to enclose a circular field having the same area? 12. What is the value of a farm, at $75 an acre, its form being a quadrilateral, with two of its opposite sides parallel, one 40 chains and the other 22 chains long, and the perpendicular distance between them 25 chains ? Ans. $5812.50. 13. What is the difference in the area of a grass plat 20 feet square and a circular plat 20 feet in diameter? 14. Find the cost, at 18 cents a square foot, of paving a space in the form of a rhombus, the sides of which are 15 ft., and a perpendicular drawn from one oblique angle will meet the opposite side 9 feet from the adjacent angle. Ans. $32.40. 15. A goat is fastened to the top of a post 4 feet high by a rope 50 ft. long. Find the circumference and area of the greatest circle over which he can graze. 16. How much larger is a square circumscribing a circle 40 rd. in diameter, than a square inscribed in the same circle ? 17. What is the value of a piece of land in the form of a triangle, whose sides are 40, 48, and 54 rods, respectively, at the rate of $125 an acre ? Ans. $724.75. 18. The radius of a circle is 5 feet; find the diameter of another circle containing 4 times the area of the first. Ans. 20 ft. 19. Find the difference in the area of a circle 36 feet in diameter, and the inscribed square. 20. How many acres in a semi-circular farm, whose radius is 100 rods? Ans. 98 A. 28 P. 21. What must be the width of a walk extending around a garden 100 feet square, to occupy one-half the ground ? 22. An irregular piece of land, containing 540 A. 36 P., is exchanged for a square piece of the same area; find the length of one of its sides. If divided into 42 equal squares, what is the length of the side of each? Ans, to last, 45.36 rd. 32 3 23. A field containing 15 A. is 30 rd. wide, and is a plane inclining in the direction of its length, one end being 120 ft. higher than the other, Find how many acres of horizontal surface it contains. 24. If a pipe 3 inches in diameter discharge 12 hogsheads of water in a certain time, what must be the diameter of a pipe which will discharge 48 hogsheads in the same time? Ans. 6 in. SOLIDS. 759. A Solid or Body has three dimensions, length, breadth, and thickness. The planes which bound it are called its faces, and their intersections, its edges. 760. A Prism is a solid whose ends are equal and parallel polygons, and its sides parallelograms. Prisms take their names from the forms of their bases, as triangular, quadrangular, pentagonal, etc. 761. The Altitude of a prism is the perpendicular distance between its bases. 762. A Parallelopipedon is a prism Parallelopipedon. bounded by six parallelogranıs, the opposite ones being parallel and equal. 763. A Cube is a parallelopipedon whose faces are all equal squares. Cube. 764. A Cylinder is a body bounded by a uniformly curved surface, its ends being equal and parallel circles. 1. A cylinder is conceived to be generated by the revolution of a rectangle about one of its sides as an axis. 2. The line joining the centres of the bases, or ends, of the cylinder is its altitude, or axis. |