THREE ROUND BODIES 501. The three round bodies here treated are of especial interest because so many familiar objects resemble them in shape. THE CYLINDER 502. The figures represent two Cylinders. The bottom and top of a cylinder are equal circles, called the bases. Its altitude is the perpendicular distance between its bases. The curved surface which extends from base to base is I called its lateral surface. A perfect cylinder is made by revolving a rectangle, as ABCD, about one of its sides, AB, as an axis. 1. What dimension of the cylinder does the axis AB measure? 2. What dimension of the base does the side AD measure? What part of the surface does AD trace? What part of the surface does the side CD trace? 3. How do you find the area of a circle? 503. Volume of a Cylinder. (Art. 177.) It is evident that if a cylinder were 1 inch high, it would contain as many cubic inches as there were square inches in the area of the base; and if it were 5 inches high, the volume would be 5 times as great (Art. 187). It follows that The volume of a cylinder is found by multiplying the measure of the area of either base by its altitude. r = That is, if a = the area of the base, c = the circumference, and the radius, we have a = 1⁄2 rc (Art. 177); or, since e is equal to 22 × 2r (Art. 175), we may write Therefore, if h is the height, a = rx 22 x 2 r, or 222. 1 the volume of a cylinder = 22 r2 × h, or wr2 × h, where the value 34, or 22, is represented by the Greek letter π (pi). WRITTEN EXERCISES 504. 1. Find the volume of a cylinder the radius of whose base is 3 ft. and whose altitude is 6 ft. 22 × 32 198, number of square feet in base. 198 × 6 = 1695, number of cubic feet in volume. 2. Find the volume of a cylinder having the radius of its base 2.5 in. and its altitude 20 in. 3. Find the number of cubic feet of water in one 14foot length of a city water main 10 in. in diameter. 4. A cylindrical standpipe is used as a reservoir. Find its capacity in cubic feet, if its height is 70 ft. and the diameter of its base 20 ft. 5. How many gallons of oil will a cylindrical tank 25 ft. high and 21 ft. in diameter hold? (Use 7 gal. = 1 cu. ft.) 505. Lateral Surface of a Cylinder. Cut a rectangular strip of paper whose width equals the altitude of some cylinder. Wrap the paper around the cylinder, and mark off the length that goes once around. Unwrap the strip and you will have a rectangle equal in area to the lateral surface. How does the length of this strip compare with the circum ference of the base? How do you find the area of the strip? From the preceding we see that — The area of the lateral surface of a cylinder is found by multiplying the circumference of its base by its altitude. WRITTEN EXERCISES 506. 1. Find the area of the lateral surface of a cylinder whose altitude is 6 ft. and the radius of whose base is 3 ft. 2 × 3 × 2 × 6 = 1134, number of square feet in area. Explain the process. 2. Find the area of the entire surface of the cylinder in problem 2, Art. 504. 3. Find how much paint will be needed for two coats on the standpipe in problem 4, Art. 504, if a gallon of paint covers 40 sq. yd. of surface with one coat. 4. A water-tube boiler has 350 tubes, each of which is 8 ft. long and 2.5 in. in diameter (inside measurement). Find in square feet the entire heating surface (interior curved surface) of the tubes. PRAC. ARITH.- - 23 THE CONE 507. The second round body is the Cone, as shown in 1. Which side of the triangle is the altitude of the cone? Which is the radius of the base? What does BC trace? 2. Which side of the triangle traces the lateral surface of the cone? 3. Make the best cone you can of stiff paper or pasteboard. Make a cylinder of the same base and altitude. Fill the cylinder with sand, and find out how many times the cone can be filled with the same amount of sand. 508. Volume of a Cone. From the preceding experiment it is found that a cone is one third of a cylinder of the same base and altitude. Therefore The volume of a cone is found by multiplying the measure of the area of its base by one third of its altitude. WRITTEN EXERCISES 509. 1. Find the volume of a cone whose height is 5 in. and the area of whose base is 12 sq. in. Find the volume of a cone whose height is 3. 6 in. and area of base 19 sq. in. in. 6. Find the volume of a cone the radius of whose base is 3 in. and whose altitude is 7 in. If the radius of the base and the altitude of a cone are given, the area of the base must first be found. (See Ex. 1, Art. 504.) Find the volumes of cones whose dimensions are 7. Radius of base 2.5 in., altitude 12 in. 8. Radius of base 28 cm., altitude .6 m. 510. Lateral Surface of a Cone. Suppose the surface of the cone shown in the figure to be cut along CA, and then unrolled and flattened out. The surface will take the form of part of a circle, CAE, in which the radius CD equals the slant height CA, and the length ADE equals the circumference of the base of the cone. By regarding a circle as made up of triangles, we found that its area equals the circumference multi 0 A plied by half the radius (Art. 177). Likewise, we find that the area of part of a circle, CAE, equals the part of the circumference, ADE, multiplied by half the radius CD. It follows that The area of the lateral surface of a cone equals the circumference of the base multiplied by half of the slant height. |