9. How many square yards are there in a circular walk, the radius, AB, of the inner edge of walk being 10 feet, and that of the outer edge, AC, being 15 feet? (Find the difference between the area of a circle of 15 ft. radius, and that of a circle of 10 ft. radius.) 10. A circular flower-bed 20 feet in diameter C B is surrounded by a walk 5 feet wide. How many square feet of surface does the walk contain? (If you have to subtract 100 times 3.1416 from 225 times 3.1416, how can you shorten the work?) 11. How many square inches are there in the surface of a frame 3 inches wide, around a looking-glass 6 inches in diameter? (Area? 3.1416.) 6 in. 12. What is the ratio between the surface of the above frame and that of the looking glass? (Indicate operations and cancel.) 13. What is the area of a walk 5 feet wide around the outside of a square plot containing 400 sq. ft.? (What is the area of the large square, including the walk?) 14. The outer edge of a walk 5 feet wide, surrounding a plot of ground, measures 120 feet, the inner edge measures 80 feet. How C many square feet does the walk contain? (The "average" length of the walk is 120 + 80 length measured on a line along the center of the walk.) 5 400 sq. ft. A ? ft. B ? ft. 5 100 ft.; that is, its 15. Find the ratio between the area of a triangle whose sides measure 16, 30, and 34 feet, respectively, and the area of another whose sides are 32, 60, and 68 feet. SURFACE OF SPHERE. 1151. Take a wooden hemisphere and drive a tack into the center of its curved surface. Commencing at the tack, carefully wind a waxed cord about the curved surface, in the way a boy winds a top. When this surface is exactly covered, cut the cord. ہم Wind the same cord around a tack driven into the plane surface of the base of the hemisphere, pressing it closely to the surface. When the latter is entirely covered, just one-half of the cord will be used. If a sphere is cut through in any direction, the section made will be a circle. The section formed when the sphere is cut through the center is called a great circle. The above experiment shows that the surface of the hemisphere is equal to that of two great circles of the same sphere. 1152. The surface of a sphere is equal to that of four great circles. Since the surface of a great circle of the sphere is diameter circumference, the surface of the sphere is diameter x circumference X 4 diameter of sphere X the circumference. Calling the radius of a circle R, and using the Greek letter = 1153. Slate Exercises. 16. Find the surface of a sphere whose radius is 1 inch. Of a sphere whose diameter is 2 inches. Of a sphere whose circumference is 6.2832 inches. 17. At 10 cents a square foot, what will be the cost of gilding a sphere 12 inches in diameter ? 18. Find the ratio between the surface of a sphere 1 foot in diameter, and the convex surface of a cylinder 1 foot high, the diameter of the base 1 foot. 19. What is the ratio between the surface of the above sphere and the entire surface of the cylinder? 20. Find the surface of a sphere whose circumference is 20 inches. CUBE ROOT. 1155. To cube a number is to employ it three times as a factor. The cube of 4, written 43, is 4 x 4 × 4, or 64. Find the cube of 1, 9, 6, 3, 5, 8, 2, 7. To find the cube root of a number is to find one of the three equal factors of the number. The cube root of 343, written 343, is 7. The cube of 25, 20+ 5, is equal to the following: We have seen (Art. 1031) that (20+ 5)2 = 202 + 2 × 20 × 5 + 52 1,200 as a trial divisor, the second number is seen to be 6 or less. Taking 5 as the second number, we add to the 1,200 three times the product of the first and second (300), and the square of the second (25), making a total of 1,525. Multiplying this sum by the second number, we get 7,625, which is equal to the difference between 15,625 and 8,000. The second number is, therefore, 5, and the cube root of 15,625 is 25. In the last example we point off three places, beginning at the right, and find the greatest cube in the first period, placing its cube root as the first figure of the answer. VOLUME OF SPHERE. 1158. Cut up a sphere (a round potato, for instance) into a number of small pieces, passing the knife in each case through the center of the sphere. Each piece is a solid, having for its base a portion of the surface of the sphere, and for its altitude the radius of the sphere. When the pieces become very numerous, the base of each may be considered a plane, and the solid a pyramid. The volume of each pyramid is equal to the base altitude; and the total volume of all, which is the volume of the sphere, is equal to the total surface of all the bases, which is the surface of the sphere, multiplied by altitude, that is, radius. therefore, Surface of sphere = 4π R2, volume of sphere = 4πR2 × R= }πR3. 1159. Slate Exercises. 1. Find the volume of a sphere whose radius is 3 inches. 2. If the diameter of a sphere is 3 inches, what is its volume? 3. What is the ratio between the volumes of two spheres whose diameters are 1 foot and 2 feet, respectively? 4. Find the ratio between the volume of a sphere 1 foot in diameter, and that of a cube whose side is 1 foot. |