EXAMPLES. 1. What is the solid contents of a block of granite 25 feet long, 4 broad, and 3 thick? Ans. 300 solid feet. 2. Find the solidity of a stick of timber 15 feet in length, 40 inches broad, and 10 inches thick. Ans. 41.5 cubic feet. To solve this question, and all similar examples by the sliding rule-First find the side of a square whose area is equal to the area of the end of the stick or prism, (as directed under example 11, ¶ 15;) then place the length in feet over 12 on D, and over the side of the equal square found on D, will be found the cubic contents in feet on the line C. Thus, place the breadth 40 inches found on C, over 40 on D, and under 10, the thickness, found on C, we find 20, the side of the equal square; then, having placed the length, 15 feet, over 12 on D, over 20 found on D we find 41.5, the cubic contents. 3. Find the number of cubic feet in a rectangular stick of timber 5 feet in length, and 35 inches broad, and 25 inches thick. Ans. 30 feet, nearly. 4. There are 5 sticks of timber, each 8 feet in length, and the ends square, one 10 inches, one 12 inches, one 17, one 19, and one 24 inches square; what is the solid contents of each? Ans. 5.54; 8; 16; 20; 32 feet. Having placed the length of the sticks, viz. 8 feet, over 12 on D, over 10 on D you will find 5.54 feet, the contents of the stick 10 inches square; and over 12 you will find 8 feet; over 17, 16 feet; over 19, 20 feet; and over 24, 32 feet. Now, since the content of any square stick, or square prism, is found on C over the side in inches on D, when the length in feet stands over 12 on D, we may reverse the case above, and find the length of a square stick which shall contain any number of feet, the side of the square end being given; thus:-Place the number of feet which the prism is required to contain, found on C, over the side of the square in inches on D, then over 12 found on D, will be found the length of the prism in feet on the line C. 5. What is the length of a square stick of timber which contains 60 cubic feet, the side of the square being 40 inches? Ans. 5.4 lineal feet. 6. How many bushels will a box hold, which is 6 feet in length, and the end 17.25 inches square? Ans. 10 bushels. To solve this example by the sliding rule, see 33, under example 3d. 7. A cubic box is 25 inches square, and it holds 30 bushels; what is its length? Ans. 8.6 feet, nearly. To solve this and like examples by the sliding rule:-Place the number of bushels on C, over the side of the square in inches on D, and over 13.387, the gauge point for bushels on D, will be found the length of the box in feet. 8. How many bushels will a box hold, which is 6 feet 6 inches long, 22 inches high, and 33 inches wide? Ans. 26.4 bushels, nearly. To solve this example by the sliding rule:-Find the side of a square box of equal capacity, as directed above; then place the length over 13.39, and over the side of the equal square box, viz. 27 inches, will be found 26.4 bushels. 9. A sleigh-box is 36 inches wide and 25 inches high, and its capacity is 40 bushels; its length is required. Ans. 8 feet, nearly. 10. How many wine gallons will a vat hold, whose dimensions are 40, 60, and 20 inches? Ans. 207.78 gallons. The gauge point for this example is 15.2 on D. 11. How many ale barrels will a vat hold, whose dimensions are 45, 60, and 40 inches? Ans. 10.66, nearly. For the gauge point, see ¶ 33, under example 6. 12. How many coal bushels of 38 quarts each, will a box hold, that is 10 feet long, 40 inches wide, and 30 high? Ans. 56.39 bushels. 13. Find the number of cubic yards in a rectangular block of sandstone, the length of which is 16 feet, its breadth 9 feet, and height 6 feet 9 inches. Ans. 36 yards. 14. The common dimensions of a brick are 8 inches in length, 4 in width, and 2 in thickness; of bricks in a cubic foot. required the number Ans. 27 bricks. 15. How many bricks of the usual size will be required to build a wall 30 feet long, a brick and a half thick, and 15 feet in height, allowing that the mortar occupies one-ninth of the space occupied by the wall? Ans. 10,800 bricks. 35. TRIANGULAR PRISM. A triangular prism is a solid comprehended under five planes, the sides being parallelograms, and the ends equal triangles. To find the solidity of any prism :— Multiply the area of the end, or its base, by its length, or height, and the product will be its cubic contents. EXAMPLES. 1. What is the solidity of a triangular prism, whose length is 10 feet 6 inches, and one side of its base 14 inches, and the perpendicular on it from the opposite angle 15 inches? Ans. 7.656 feet. 2. There is a triangular prism, the ends of which are equilateral triangles 30 inches on a side, and the length of the prism is 24 feet; required its contents in feet and in bushels. Answers, 65 feet, and 52.2 bushels, nearly. To find the number of feet in the equilateral triangular prism: -Place its length in feet and decimal parts of a foot over 18.234 on D, and over the side (30 inches) found on D, will be found its solidity in cubic feet on C. See19, under examples 5, 6, and 7. To find the gauge point for bushels in an equilateral triangular prism:-Divide of 2150.42 by .4330127, and extract the square root of the quotient; and said root, viz. 20.345, will be the required gauge point, when the length of the prism is given in feet. Consequently, place the length in feet, viz. 24 on C, over 20.345 on D, and over the side of the base in inches found on D, will be found the contents in bushels on C. 3. The side of the base of an equilateral triangular prism is 120 inches; required its length, its solidity being 100 cubic feet. Ans. 2.3094 feet. 4. What is the solidity of a rhombic prism 18 feet in length, the side of the base being 26 inches, and the perpendicular distance between the parallel sides 15 inches? Ans. 48.5 cubic feet. 5. What is the content of a rhomboidal prism 40 feet in length, the longest edges of the base being 38 inches, and the perpendicular between them 7.6 inches? Ans. 80.4 cubic feet. See the ends of the rhombic and rhomboidal prisms in paragraphs 16 and 17. See 34, under example 2d. 6. What is the solidity of a regular pentagonal prism 7 feet in height, the edge or side of the base being 18 inches? Ans. 27.1 cubic feet. The gauge points for the pentagon, hexagon, and octagon, may be found on the last page of 21. In the above example, place the altitude of the prism over 9.15 on D, and over the side in inches found on D, will be found the contents in feet on C. 7. What is the solidity of a regular octagonal prism, whose altitude is 10 feet, and the edge of its base 30 inches? and how many bushels will it contain? Answers, 301.794 cubic feet; and 242 bushels and 2 pecks, nearly. The gauge point for feet is 5.46, and for bushels 6.092 on D. 8. The edge of the base of an octagonal prism is 20 inches, and its contents equal to 40 bushels; required its length. Ans. 3.72 feet, nearly. To solve the last example by the sliding rule:-Place 40 on C over 20 on D, and over the gauge point for bushels, viz. 6.092, will be found the length or altitude of the prism. 9. What is the solidity of a regular hexagonal prism, whose altitude is 5 feet, and the edge of its base 9 inches? Ans. 7.28 feet, nearly. The gauge point is 7.45 inches on D. To find the surface of any prism :— Multiply its perimeter by the length of its lateral edge, and the product will be its lateral surface, to which add double the area of the base, and the sum will be the whole surface of the solid. EXAMPLES. 1. What is the surface of an oblique prism 26 feet long, the perimeter of a section perpendicular to one of its lateral edges being 19 feet, and its base a rectangle 6 feet long and 4 broad? Ans. 542 square feet. 2. What is the surface of a right rectangular parallelopiped, whose length is 36 feet, breadth 10 feet, and thickness 8 feet? Ans. 1456 square feet. 3. The length of a rectangular cistern within is 3 feet 2 inches, the breadth 2 feet 8 inches, and height 2 feet 6 inches; required the internal surface, and the expense of lining it with lead at 4 cents per lb., the lead being 7 lbs. weight per square foot. Ans. 371 square feet, and $10.53. 4. What is the surface of a right triangular prism, its length being 20 feet, and the sides of its base 6, 8, and 10 feet? Ans. 528 feet. 5. What is the surface of a regular pentagonal prism, whose length is 32 feet, and a side of the base 61 feet? Ans. 1150.037 feet. |