473. To find the diameter or circumference of a circle, when the area is given. 1. What is the diameter of a circle whose area is 1319.472 ? 1319.472 SOLUTION. 1680; √/1630 = 40.987+, the diameter 2. What is the circumference of a circle whose area is 19.635? 19.635 SOLUTION. 3.1416 FORMULAS: 1. 2. = = 6.25; 1/6.25 = 2.5, radius; 15.708, the circumference. 3. The area of a circular lot is 38.4846 sq. rd. Find its diameter. 4. The area of a circle is 78.75 sq. yd. Find its diameter and its circumference. 5. The area of a circle is 286.488 square feet. Required the diameter and the circumference. SOLIDS.* 474. A prism is a solid whose bases are equal and parallel polygons, and whose sides are parallelograms. The planes which bound a solid are called faces, and their intersections edges. The altitude of a prism is the perpendicular distance between its bases. A prism is triangular, quadrangalar, pentagonal, etc., according as its bases have three sides, four sides, five sides, etc. *For some other definitions, see page 162. Triangular Prism. Quadrangular Prism. Pentagonal Prism Hexagonal Prism. 475. A cylinder is a body bounded by a uniformly curved surface, its ends being equal and parallel circles. The altitude or axis of a cylinder is the line joining the centers of the bases or ends. 476. A pyramid is a body having for its base a polygon, and for its other faces three or more triangles, which terminate in a common point called the vertex. Cylinder. Pyramid. 477. A cone is a body having a circular base, and whose convex surface tapers uniformly to the vertex. Cone. The altitude of a pyramid or of a cone is the perpendicular distance from its vertex to the plane of its base. The slant height of a pyramid is the perpendicular distance from its vertex to one of the sides of the base; of a cone, is a straight line from the vertex to the circumf. of the base. The frustum of a pyramid or cone is that part which remains after cutting off the top by a plane parallel to the base. Frustums. 478. A sphere is a body bounded by a uniformly curved face, all the points of which are equally distant from a point within called the center. The diameter of a sphere is a straight line passing through the center of the sphere, and terminated at both ends by its surface. The radius of a sphere is a straight line drawn from the center to any point in the surface. PROBLEMS. 479. To find the convex surf. of a prism or cylinder. 1. Find the area of the convex sur face of a prism whose altitude is 7 ft., and its base a pentagon, each side of which is 4 feet. SOLUTION.-4 ft. x 5 =20 ft., perimeter. 20 ft. x 7 = 140 sq. ft., convex surface. 2. Find the area of the convex surface of a triangular prism, whose altitude is 8 feet, and the sides of its base 4, 5, and 6 feet, respectively. SOLUTION.-4 ft +5 ft.+6 ft. = 15 ft., pe rimeter. 15 ft x81 = 127 sq. ft., convex surface. 3. Find the area of the convex surface of a cylinder whose altitude is 2 ft. 5 in., and the circumference of its base 4 ft. 9 in. SOLUTION.-2 ft. 5 in. = 29 in.; 4 ft. 9 in. = 57 in. 57 in. x 29-1653 sq. in. = 11 sq. ft. 69 sq. inches, convex surface. FORMULA: Perimeter of base x altitude = convex surface. To find the entire surface, add the area of the bases or ends. 4. If a gate 8 ft. high and 6 ft. wide revolve upon a point in its center, what is the entire surface of the cylinder described by it? 5. Find the superficial contents, or entire surface of a prism 8 ft. 9 in. long, 4 ft. 8 in. wide, and 3 ft. 3 in. high. 6. Find the entire surface of a cylinder formed by the revolution about one of its sides of a rectangle that is 6 ft. 6 in. long and 4 ft. wide? 7. Find the entire surface of a prism whose base is an equilateral triangle, the perimeter being 18 ft., and the altitude 15 feet? 480. To find the volume of any prism or cylinder. 1. Find the volume of a triangular prism, whose altitude is 20 ft., and each side of the base 4 ft. SOLUTION.-The area of the base is 6.928 sq. ft. (465). 6.928 sq. ft. x 20 = 138.56 cu. ft., volume. 2. Find the volume of a cylinder whose altitude is 8 ft. 6 in., and the diameter of its base 3 ft. SOLUTION.-32.78547.0686 sq. ft., area of base (472). 7.0686 sq. ft. x 8.5 = 60.083 cu. ft., volume. FORMULA: Area of base × altitude = volume. 3. What is the volume of a log 18 ft. long and 14 ft. in diameter ? 4. Find the solid contents of a cube whose edges are 6 ft. 6 in. ? 5. Find the cost of a piece of timber 18 in. square and 40 ft. long, at $.30 a cubic foot. 6. What is the value of a log 24 ft. long, of the average circumference of 7.9 ft., at $.45 a cubic foot? 481. To find the convex surface of a pyramid or cone. 1. Find the convex surface of a triangular pyramid, the slant height being 16 ft., and each side of the base 5 feet. SOLUTION. (5 ft.+5 ft. +5 ft.) x 16÷2 = 120 sq. ft., convex surface. 2. Find the convex surface of a cone whose diameter is 17 ft. 6 in., and the slant height 30 ft. SOLUTION.-17.5 ft. x 3.1416 = 54.978 ft. x 30÷2 = 54.978 ft., circumference. 324.67 sq. ft., convex surface. FORMULA: Perim. of base × slant height = convex surf. To find the entire surface, add to this product the area of the base. 3. Find the entire surface of a pyramid whose base is 8 ft. 6 in. square, and its slant height 21 feet. 4. Find the entire surface of a cone the diameter of whose base is 6 ft. 9 in. and the slant height 45 feet. 482. To find the volume of any pyramid or cone. 1. What is the volume, or solid contents, of a square pyramid whose base is 6 ft. on each side, and its altitude 12 ft. SOLUTION.-6 × 6 × 12÷3 = 144 cu. ft., volume. 2. Find the volume of a cone, the diameter of whose base is 5 ft. and its altitude 10 ft. SOLUTION.—52 ft. × .7854°× 101÷3 = 68.721 cu. ft., volume. FORMULA: Area of base ×§ altitude = volume. 3. Find the solid contents of a cone whose altitude is 24 ft., and the diameter of its base 30 inches. 4. What is the cost of a triangular pyramid of marble, whose altitude is 9 feet, each side of the base being 3 feet, at $2 per cubic foot? 5. Find the volume and the entire surface of a pyramid whose base is a rectangle 80 ft. by 60 ft., and the edges which meet at the vertex are 130 feet. |