399. A prism is a figure whose ends are equal plane figures, and whose faces are parallelograms. The sum of the sides which bound the base is called the perimeter of the base; and the sum of the parallelograms which bound the solid, is called the convex surface. 400. To find the convex surface of a right prism. Rule.-Multiply the perimeter of the base by the perpendicular height, and the product will be the convex surface. (Bk. VII. Prop. I.) Examples. 1. What is the convex surface of a prism whose base is bounded by five equal sides, each of which is 35 feet, the altitude being 26 feet? 2. What is the convex surface when there are eight equal sides, each 15 feet in length, and the altitude is 12 feet? 401. To find the solid contents of a prism. Rule.-Multiply the area of the base by the altitude, and the product will be the contents. (Bk. VII., Prop. XIV.) Examples. 1. What are the contents of a square prism, each side of the square which forms the base being 15, and the altitude of the prism 20 feet? 2. What are the contents of a cube, each side of which is 24 inches? 3. How many cubic feet in a block of marble, of which the length is 3 feet 2 inches, breadth 2 feet 8 inches, and height or thickness 2 feet 6 inches? 4. How many gallons of water will a cistern contain, whose dimensions are the same as in the last example? 5. Required the contents of a triangular prism whose height is 10 feet, and area of the base 350? 402. A cylinder is a figure with circular ends. The line EF is called the axis, or altitude; and the circular surface, the convex surface of the cylinder. 403. To find the convex surface. Rule.-Multiply the circumference of the base by the altitude, and the product will be the convex surface. (Bk. VIII., Prop. I.) Examples. 1. What is the convex surface of a cylinder, the diameter of whose base is 20, and the altitude 50? 2. What is the convex surface of a cylinder, whose altitude is 14 feet, and the circumference of its base 8 feet 4 inches? 3. What is the convex surface of a cylinder, the diameter of whose base is 30 inches, and altitude 5 feet? 404. To find the contents of a cylinder. Rule.-Multiply the area of the base by the altitude: the product will be the contents. (Bk. VIII., Prop. II.) Examples. 1. Required the contents of a eylinder, of which the altitude is 12 feet, and the diameter of the base 15 feet? 2. What are the contents of a cylinder, the diameter of whose base is 20, and the altitude 29? 3. How many barrels of wine will a cylindrical vat fill, the diameter of whose base is 12, and the altitude 30? 4. What are the contents, in hogsheads, of a cylindrical cistern, the diameter of whose base is 16, and altitude 9? 5. What are the contents of a cylinder, the diameter of whose base is 50, and altitude 15? 405. A pyramid is a figure formed by several triangular planes united at the same point S, and terminating in the different sides of a plain figure, as ABCDE. The altitude of the pyramid is the line S O, drawn perpendicular to the base. D B Ꭺ 406. To find the contents of a pyramid. Rule.-Multiply the area of the base by one-third of the altitude. (Bk. VII., Prop. XVII.) Examples. 1. Required the contents of a pyramid, of which the area of the base is 95, and the altitude 15? 2. What are the contents of a pyramid, the area of whose base is 260, and the altitude 24 ? 3. What are the contents of a pyramid, the area of whose hase is 207, and altitude 18? 4. What are the contents of a pyramid, the area of whose base is 403, and altitude 36? 5. What are the contents of a pyramid, the area of whose base is 270, and altitude 16? 6. A pyramid has a rectangular base, the sides of which are 25 and 12: the altitude of the pyramid is 36: what are its contents? 407. A cone is a figure with a circular base, and tapering to a point called the vertex. The point C is the vertex, and the line CB is called the axis, or altitude. C B 408. To find the contents of a cone. Rule.-Multiply the area of the base by the altitude, and divide the product by 3. (Bk. VIII., Prop. V.) Examples. 1. Required the contents of a cone, the diameter of whose base is 5, and the altitude 10? 2. What are the contents of a cone, the diameter of whose base is 18, and the altitude 27? 3. What are the contents of a cone, the diameter of whose base is 20, and the altitude 30? 4. What are the contents of a cone, whose altitude is 27 feet, and the diameter of the base 10 feet? 5. What are the contents of a cone, whose altitude is 12 feet, and the diameter of its base 15 feet? GAUGING. 409. GAUGING is a process for determining the capacity or contents of casks. The mean diameter of a cask is found by adding to the head diameter, two-thirds of the difference between the bung and head diameters, or, if the staves are not much curved, by adding six-tenths. This reduces the cask to a cylinder. Then, to find the contents, we multiply the square of the mean diameter by the decimal .7854, and the product by the length. This will give the contents in cubic inches. Then, if we divide by 231, we have the contents in gallons (Art. 199). Multiply the length by the square of the mean diameter, then by the decimal .7854, and divide by 231. OPERATION. 7x d2 × ·7854 231 = 1 x d2 x .0034. If, then, we divide the decimal .7854 by 231, the quotient, carried to four places of decimals, is .0034, and this decimal multiplied by the square of the mean diameter and by the length of the cask, will give the contents in gallons. 410. Hence, for gauging or measuring casks, we have the following Rule.-Multiply the length by the square of the mean diameter; then multiply by 34, and point off four decimal places, and the product will then express gallons and the decimals of a gallon. Examples. 1. How many gallons in a cask whose bung diameter is 36 inches, head diameter 30 inches, and length 50 inches? We first find the difference of the diameters, of which we take two-thirds, and add to the head diameter. We then multiply the square of the mean diameter, the length, and 34 together, and point off four decimal places in the product. 36 OPERATION. 30= of 6 6 = 4 30+ 4 = 34 342 1156 1156 × 50 × 34 = 196.52 gal. 2. What is the number of gallons in a cask whose bung diameter is 38 inches, head diameter 32 inches, and length 42 inches? 3. How many gallons in a cask whose length is 36 inches, bung diameter 35 inches, and head diameter 30 inches? 4. How many gallons in a cask whose length is 40 inches, head diameter 34 inches, and bung diameter 38 inches? 5. A water-tub holds 147 gallons; the pipe usually brings in 14 gallons in 9 minutes; the tap discharges at a medium, 40 gallons in 31 minutes. Now, supposing these to be left open, and the water to be turned on at 2 o'clock in the morning; a servant at 5 shuts the tap, and is solicitous to know at what time the tub will be filled, in case the water continues to flow. |