29. The average diameter of the earth is 7918 mi. How many square miles are there in its surface? 30. Calling s the surface and r the radius of a sphere, write an equation showing the relation between s and r. It is seen on p. 169 that the volume of a right triangular prism equals the volume of a square prism whose base and altitude are equal to the base and the altitude of the square prism. س 4 6 FIGURE 1 FIGURE 2 31. Find the volume of a right triangular prism whose altitude is 18" and whose base is a triangle having a base of 8" and an altitude of 5′′ inches. If a triangular pyramid be carefully modeled (Fig. 1) and filled with sand it will be found that just 3 times the volume of the pyramid is equal to the volume of the model of a triangular prism (see Fig. 2) of equal base and equal altitude. It is proved in geometry that the volume of any pyramid equals of the volume of a prism having an equal base and an equal altitude. Notice in Fig. 2 how a triangular prism may be completed on a triangular pyramid having the same base and the same altitude as the prism. 32. Calling V the volume, B the area of the base, and a the altitude of a triangular pyramid, write an equation, showing the way V would be computed from B and a. 33. Find the volume of a pyramid whose base contains 16 sq. in. and whose altitude is 12 inches. 34. The Great Pyramid of Egypt is 481 ft. high and its base is a 756' square. If it were solid and had smooth faces, how many cubic feet of masonry would it contain? 35. At the close of the nineteenth century the United States had 195,887 mi. of railroad. How many times would these railroads, if placed end to end, encircle the earth? (Use π 34, and the radius of the earth = 3959 Iniles.) = 36. The ties used for these roads would contain wood enough to make a pyramid 1395' high with a 2192' square for its base. How many cubic feet of wood were used for the railroad ties? 37. It has been computed that the materials used for the road beds for these railroads would make a solid pyramid 2470 ft. high and having a 3870 ft. square for its base. How many cubic feet would this make? If the entire surface of a globe, or sphere, were divided up into small triangles and rectangles, and the sphere were cut up by planes cutting along the curved sides of these figures and passing through the center, O, the volume of the sphere would be divided up approximately into small pyramids, having their vertices at the center (see Fig. 1). The volume of each pyramid would be the product of its base by of the radius of the sphere. The sum of the FIGURE 1 areas of the bases of all these pyramids would equal the surface of the whole sphere and the sum of the volumes of all the pyramids would equal the surface of the whole sphere multiplied by of the radius of the sphere. 38. Calling V the volume and r the radius of a sphere, give the meaning of the formula: 39. Find the number of cubic inches in a sphere of 2′′ radius. 40. How many cubic inches in a croquet ball 4" in diameter? In a tennis ball 1.75" in diameter? In a baseball 2.9" in diameter? In a globe 10.15" in diameter? 41. Calling the sun, the moon, and the planets all spheres with diameters in miles as in the following table, compute two or three of their circumferences in miles, their surfaces in square miles and their volumes in cubic miles: 42. What is the ratio of the surface (including the ends) of a cylinder having an altitude 12 inches and a radius of 6 inches, to the surface of a sphere 12 inches in diameter? What is the ratio excluding the end surfaces? 43. What is the ratio of the areas of the curved surfaces of a right cone and a right cylinder, both having an altitude 12 inches and radius of base of 6 inches? 44. Find the ratio of the volume of a sphere 12 inches in diameter to the volume of a right cylinder 12 inches high and 12 inches in diameter. 45. Find the ratio of volumes of a right cone 12 inches high and having a circular base 12 inches in diameter, and a sphere 12 inches in diameter. 46. Find the ratio of volumes of a cone and a cylinder each 12 inches high and each having a circular base 12 inches in diameter. $117. The Right Triangle. A right triangle is a triangle that has a right angle. EXERCISE I. To find the relation of the squares of the sides of an isosceles right triangle. Construct an isosceles right triangle and on each of its three sides draw a square. Draw the dotted lines and cut the side squares as shown in Fig. 1. Fit the pieces over the large square on the hypothenuse. If the area of each side square were 9 sq. in. what would be the area of the large square? FIGURE 1 EXERCISE II.-To find the relation of the squares of the sides of a triangle, having one of the shorter sides twice as long as the other. 3 2 4 2 5 4 With a square-cornered card draw squares on each of the three sides. Cut off the side squares. Cut the larger side-square first into two equal rectangles, and then cut each rectangle along diagonals, as shown. Place the several pieces as the numbers in Fig. 2 indicate. It is plain that the area of the square on the hypothenuse equals the sum of the areas on the side-squares, as above. EXERCISE III.-To find the relation of the squares of the sides of any right triangle. Draw any right triangle and then draw squares on each of the sides. dicular to the parallel. Cut as shown, and place the pieces on the largest square as shown by the numbers in Fig. 1. Thus, it is clear that with any right triangle the sum of the squares on the sides equals the square on the hypothenuse. 1. Denoting the length of either short side of Fig. 1, p. 217, by a, what denotes the area of the square drawn upon this side? 2. Denoting the length of the hypothenuse of Fig. 1, p. 217, by h, what denotes the area of the square drawn on the hypothenuse? 3. Write an equation from Fig. 2, p. 217, showing the relation between a and h2. 4. Denote the lengths of the three sides of any right triangle (Fig. 1) by a, b, and h (h being the hypothenuse). What will denote the areas of each of the squares on the three sides? 5. Write an equation showing the relation of the squares of the sides of any right triangle. 6. The sides of a right triangle are 3" and 4". What is the length of the hypothenuse? |