61 PART III. OF VOLUME. PROBLEM I. To find the volume of a prism or cylinder. RULE. Multiply the base by the height, and the product will be the number of cubes (that is, solid squares) contained in the body. Note. The length, breadth, and height, must always be expressed in the same sort of measure, whether it be yards, feet or inches; if they are not so expressed in the proposition, they must be reduced before any thing else is done. Example 1. A wall is 2,8 yards high; 0,6 thick ; and 104,5 yards long; how many cubick feet does it contain ? 2,8 multiplied by 0,6-gives 1,68 square yds. 1,68 multiplied by 104,5-gives 175,560 square yds. 2. A pile of wood in the form of a parallelopiped, is 54,8 feet long; 22,3 feet wide; and 37,1 feet in height; how many cubick feet does it contain? Multiply these three numbers together, and the answer will be 45337,684 cubick feet. 3. A cylindrical caldron is 8,3 feet deep, and 13 feet wide; what is its capacity (that is, how many cubick feet will it contain?) The width or diameter is 13, the radius must be 6,5. Multiply 6,5 by 6,5 and the product by 34 (Part II. prob) and you have the superficies of the base, which multiply by the height 8,3 and you have the capacity required. 7 Ans. 1102,124 see feet. zubiak 4. A common brick is generally 8 inches long, 4 wide, and 2 thick. What is its volume? How many will it take to make a cubick foot of masonry? Vol. 64 sq.in 27 bichsen l.ft. How many bricks will it take to construct a wall, 300 feet long, 6 feet high, and 1,5 thick? 5. Ans. 72900. Note. It will be seen that this and the preceding calculation, make no allowance for mortar. 6. A well is 6,9 yards deep, and 1,2 yards in diameter. I wish to make a wall in it, 0,4 yards thick ; how many cubick feet of stone will build it? Calculate the well as if it were to be entirely filled up. Its diameter being 1,2 it radius must be half that, or 0,6 tenths; to this add, 0,4 tenths, the proposed thickness of the wall, and you have 10 tenths, or a whole yard, for the radius of the well. Then subtract the empty part of the well, which forms another cylinder, whose radius is 0,6 tenths, as above mentioned. 1 multiplied by 1 and by 3 gives 3,14 the base of the the answer. PROBLEM II. To gauge a cask. RULE. Take the superficies of the base, and twice that of the centre at the bung hole, (Prob. 3, Part 1.) d the two amounts, and multiply the product by a third of the length. Note. All these measurements should be of the inside or clear, otherwise the thickness of the wood will be included. Example 1. A cask is 31 inches in diameter at the bung, 28 at the base or head, and its length is 54 inches. What is its capacity? Amount, neglecting fractions, 2126 Multiply this by 18, which is a third of the length, and you have the answer, 38268 cubick inches. There are 231 cubick inches in a gallon. To find then how many gallons the above cask contains, divide its contents by 231. 2. Required the contents of a cask whose diameter at the bung is 38,6 inches, at the head or base 33,4 and whose length is 63,9 inches. Ans? 68540, 783 inches. 3. Required the contents of a cask whose circumference at the bung is 90 inches, at the base 80 inches, and whose length is 48 inches. s? 284115 cubick wishes 1229 Gallons. PROBLEM III. To find the volume or solid contents of a pyramid or a cone. RULE. Multiply the base by the height, and take a third of the product. Example 1. A loaf of sugar has a base 4,8 inches in diameter, and is 12,3 inches in height. What are its contents? & diameter the radius or half Find the contents of the base by multiplying 24 by 24 and the product by 3. Multiply the latter product by the height 12,3 and divide the product by 3 to find a third of it, which will be the answer. Ans.74, 21 Cubine the envir 2. The base of a pyramid is a pentagon of equal sides, (2d Class, fig. 12.) each side being 14 inches. The height is 22 inches. What are its solid contents? 3. The base of a pyramid is a right angled triangle, (1st Class, fig. 9.) of which the base, or longest side, is 13 inches, the shortest 6. The height of the pyramid is 19 inches. Required the solid contents. Find the superficies of the base by Prob. I, Part I. PROBLEM IV. To find the volume or solid contents of a truncated cone of parallel bases. Note. A truncated cone is one whose top is cut off. RULE. Multiply the radius of each base by itself, and multiply them together. Add together the three products. Multiply the whole sum by the height, and add to this product a third of a ninth of it, (that is, a 27th.) Example 1. A bucket is 14,5 inches in diameter at top, and 11,2 at bottom. Its perpendicular height is 17,5 inches. Required its solid contents. 14,5 multiplied by 14,5 gives 11,2 multiplied by 11,2 gives 14,5 multiplied by 11,2 gives 210,25 125,44 162,40 498,09 8715 968,3 322,7 498 multiplied by the height 17,5 gives A ninth of which is . And a third of the ninth is. Cubick inches, 9037,7 2. How many such buckets of water would it take to fill the caldron mentioned in Example 3, Prob. I. of this Part. Part END. |