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Session 12

Session 12. VOLUME AND CUBIC MEASUREMENT. Volume and Cubic Measurement. Volume or Cubic measure refers to measurement of the space occupied by a body. Each body has three linear dimensions: length, height, and depth.

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Session 12

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  1. Session 12 VOLUME AND CUBIC MEASUREMENT

  2. Volume and Cubic Measurement Volume or Cubic measure refers to measurement of the space occupied by a body. Each body has three linear dimensions: length, height, and depth. The principles of volume measure are applied in this unit to three common shapes and the combination of these three shapes: (1) the cube, (2) the rectangular solid, and (3) the cylinder.

  3. The Concept of Volume Measure Volume measure is the product of three linear measurements. Each measurement must be in the same linear unit. The product is called the volume of the solid or body. Volume is expressed in cubic units.

  4. The Concept of Volume Measure The standard unit of volume or cubic measure is the cubic inch. The cube on the left is 1 in. long,1 in. high, and 1 in. deep. The cube on the right is 1 ft long, 1 ft high, and 1 ft deep and measures 1 ft3. The cube on the right is 12 in. long, 12 in. high, and 12 in deep and measures 1728 in3.

  5. The Concept of Volume Measure One cubic foot is the space occupied by a cubical body that is 1’ long by 1’ high by 1’ deep One cubic yard is the space occupied by a cube that is 1 yd long, 1 yard high, and 1 yard deep.

  6. The Concept of Volume Measure When expressing a cubic measurement, you can do it several different ways. A cube measuring 6 inches a side can be expressed the following ways 216 inches cubed or 216 cubic inches 216 in³

  7. The Concept of Volume Measure

  8. Expressing Units of Volume Measure A volume in cubic inches may be expressed in cubic feet by dividing by 1,728 (1,728 cubic inch = 1 cubic foot). That is, small to big, divide. Volumes given in cubic feet may be expressed in cubic yards by dividing by 27 (27 cubic feet = 1 cubic yard). Again, small to big, divide.

  9. Rule for Expressing Unit of Volume Measure as a larger unit Divide the given volume by the number of cubic units contained in the required larger units. Express the quotient in terms of the required larger cubic units.

  10. Expressing Unit of Volume Measure as a larger Unit - Example Express 5,184 cubic inch as cubic feet. Small to big, divide Divide the given volume (5,184) by the number of cubic inches contained in one cubic foot (1,728). Express the quotient in terms of the required larger cubic units.

  11. Expressing Unit of Volume Measure as a larger Unit - Example Express cubic ft as cubic yards Small to big, divide 3456 cu ft = cu yd. 1 cu yd = 27 cu ft 3456 cu ft  27 cu ft/ cu yd = 128 cu yd

  12. Rule for Expressing a larger Unit of Volume Measure as a smaller unit Multiply the given unit of volume by the number of smaller cubic units contained in one of the required smaller units. Express the product in terms of the required smaller cubic units.

  13. Expressing a larger Unit of Volume Measure as a smaller unit - Example Express 10 cubic yards in cubic feet. Big to small, multiply Multiply the given volume (10) by the number of cubic feet contained in one cubic yard (27). Express the product (270) in terms of the required smaller cubic units.

  14. Expressing a larger Unit of Volume Measure as a smaller unit - Example Express cubic feet as cubic inches Big to small, multiply 3⅝ cu ft = cu in. 1 cu ft = 1728 cu in 1728 cu in/ cu ft x 3⅝ cu ft = 6,264 cu in

  15. EXERCISE 8–7 Express each volume as indicated. Round each answer to the same number of significant digits as the original quantity. 2. 860 cu in = ? cu ft 0.4977 cu ft, 0.5 cu ft 4. 187 cu ft = ? cu yd 6.92593 cu yd, 6.93 cu yd 6. 18,000 cu in = ? cu yd 0.3858 cu yd, 0.39 cu yd 8. 124.7 cu ft = ? cu yd 4.619 cu yd, 4.619 cu yd 10. 51,000 cu in ? cu yd 1.09 cu yd, 1.1 cu yd

  16. EXERCISE 8–7 Express each volume as indicated. Round each answer to the same number of significant digits as the original quantity. 12. 0.325 cu ft ? cu in 561.6 cu in, 562 cu in. 6561 cu ft, 6561 cu ft 14. 243.0 cu yd ? cu ft 16. 0.09 cu yd ? cu in 4199.04 cu in, 4000 cu in 18. 0.36 cu yd ? cu ft 9.72 cu ft, 9.7 cu ft 20. 0.1300 cu yd ? cu in 6065.28 cu in, 6065 cu in

  17. Volumes of Prisms and Cylinders The ability to compute volumes of prisms and cylinders is required in various occupations. Heating and air-conditioning technicians compute the volume of air in a building when determining heating and cooling system requirements. The displacement of an automobile engine is based on the volume of its cylinders.

  18. POLYHEDRONS A polyhedron is a three-dimensional (solid) figure whose surfaces are polygons. In practical work, perhaps the most widely used solid is the prism. A prism is a polyhedron that has two identical (congruent) parallel polygon faces called bases and parallel lateral edges. The other sides or faces of a prism are parallelograms called lateral faces.

  19. PRISMS A lateral edge is the line segment where two lateral faces meet. An altitude of a prism is a perpendicular segment that joins the planes of the two bases. The height of the prism is the length of an altitude.

  20. PRISMS Prisms are named according to the shape of their bases, such as triangular, rectangular, pentagonal, hexagonal, and octagonal.

  21. PRISMS Some common prisms are shown.

  22. VOLUMES OF PRISMS The volume of any prism (right or oblique) is equal to the product of the base area and altitude. V = ABh where V = volume AB = area of base h = height

  23. the Cube AS A prism If the original surface is an square and its face is extended to add depth, the resulting figure is a solid. When all the corners are square and all lengths are equal, it is called a cube or cubical solid.

  24. Rule for computing the Volume of a CubIC PRISM Express the dimensions for length, depth, and height in the same linear unit of measure when needed. Multiply the area of base x altitude or height. Express the product in terms of units of volume measure. Express the resulting product, if needed, in lowest terms.

  25. Computing the Volume of a Cube Find the volume of a cube, each side of which is 8 inches long Multiply the area (length times depth) by the altitude or height. Express the product (512) in terms of volume measure.

  26. Computing the Volume of a Cube Find the volume in cubic inches of a cube that measures 1’ –9” on a side. Express 1’ – 9” as 21” Multiply the area of base (length x depth) x height. Express the product (9261) in terms of volume.

  27. Volume of a righT Rectangular prism A Right Rectangular Prism resembles a cube except that the faces or sides are rectangular in shape. The volume of a rectangular solid is equal to the area of base x height.

  28. Rule for finding the Volume of a RIGHT Rectangular PRISM Express the dimensions for length, depth, and height in the same linear unit of measure when needed. Multiply the area of base x height. Express the product in terms of units of volume measure. Express the resulting product, if needed, in lowest terms.

  29. Computing the Volume of a RIGHT Rectangular PRISM Find the volume the block. Express all units the same as inch Multiply the area of base by the height (9”). Express the product (2880) in terms of volume measure.

  30. EXAMPLE 2. A concrete slab with a rectangular base as shown. a. Find the number of cubic yards of concrete required. 1. Find the area of the rectangular base. AB = 40.5 ft x 20.0 ft = 810 ft2 2. Find the volume of the slab. V = 810 ft2 x 0.50 ft = 405 ft3 3. Find the volume in cubic yards. 405 ft3÷ 27 ft3 /yd3 = 15 yd3

  31. EXAMPLE 2. A concrete slab with a rectangular base as shown. One cubic yard of concrete weighs 3,700 pounds. Find the weight of the slab 4. Find the weight of the slab. 15 yd3 x 3,700 lb/yd3 = 56,000 lb

  32. EXERCISE 28–2 2. Compute the volume of a prism with a height of 26.500 cm and a base of 610.00 cm2. 26.500 cm x 610.00 cm2 = 16,165 cm3

  33. EXERCISE 28–2 4. Find the capacity in gallons of a rectangular tank with a base area of 325.0 square feet and a height of 12.8 feet. Round the answer to 2 significant digits. 325.0 ft2 x 12.8 ft = 4,160 ft3 4,160 ft3 x 7.5 gal/ft3 = 31,000 gal

  34. EXERCISE 28–2 6. How many cubic feet of air are heated in a room 28’6” long, 22’0” wide, and 8’6” high? Round the answer to 2 significant digits. 28.5’ x 22’ x 8.5’ = 5300 ft3

  35. EXERCISE 28–2 8. An excavation for a building foundation is 60.0 feet long, 35.0 feet wide, and 14.0 feet deep. • How many cubic yards of soil are removed? Round the answer to 3 significant digits. b. How many truck loads are required to haul the soil from the building site if the average truck load is 3.5 cubic yards? Round the answer to 2 significant digits. 60.0’ x 35.0’ x 14.0’ 29,400 ft3 29,400 ft3 ÷ 27 ft3/ yd3 1,089 yd3 1,089 yd3 ÷ 3.5 yd3/truckload 310 truckloads

  36. CYLINDERS Cylinders are used in many industrial and construction applications. Pipes, shafts, support columns, and tanks are a few of the practical uses made of cylinders. A circular cylinder is a solid that has identical (congruent) circular parallel bases. The surface between the bases is called lateral surface.

  37. CYLINDERS The altitude of a circular cylinder is a perpendicular segment that joins the planes of the bases. The height of a cylinder is the length of an altitude. The axis of a circular cylinder is a line that connects the centers of the bases.

  38. VOLUMES OF CYLINDERS As with a prism, a right circular cylinder has a uniform cross-section area. The formula for computing volumes of right circular cylinders is the same as that of prisms. The volume of a cylinder is equal to the product of the base area and height. V = ABh where V = volume AB = area of base (r2 or 0.7854d2) h = height

  39. Application of Volume Measure to Cylinders The volume of a cylinder is the number of cubic units that it contains. The volume of a cylinder is found by multiplying the area of the base times the length or height.

  40. Rule for finding the Volume of a Cylinder Express the dimensions for length, depth, and height in the same linear unit of measure when needed. Compute the area of the base. Multiply the area by the height or length of cylinder. Express the product in terms of units of volume measure. Express the resulting product, if needed, in lowest terms.

  41. EXAMPLE 1. Find the volume of a cylinder with a base area of 30.0 square centimeters and a height of 6.0 centimeters. Compute area of base (.7854 x d²) Multiply the area of base (706.86) by the height (6.0).

  42. EXAMPLE 2. A cylindrical tank has a 6’3.0” diameter and a 5’9.0” height. a. Find the volume of the tank. Compute area of base (.7854 x d²) Multiply the area of base (30.6797) by the height (5.75).

  43. EXAMPLE 2. A cylindrical tank has a 6’3.0” diameter and a 5’9.0” height. b. Find the capacity in gallons. 176.4 cu ft x 7.5 gal/cu ft

  44. EXAMPLE 3. A length of pipe is shown. Find the volume of metal in the pipe. Find the area of outside circle (OD). (.7854 x d²) Find the area of the hole (ID). (.7854 x d²) Find the cross-sectional area. (OD - ID) Find the volume. Base (3.487176) x the height (50.0).

  45. EXERCISE 28–4 2. Compute the volume of a right circular cylinder with an altitude of 0.40 meter and a base area of 0.30 square meter. 0.40 cm x 0.30 m2 = 0.12 m3

  46. EXERCISE 28–4 4. Compute the weight of a cedar post that has a base area of 20.60 square inches and a length of 8.25 feet. Cedar weighs 23.0 pounds per cubic foot. Round the answer to 3 significant digits. 20.60 in2 ÷ 144 in2/ft2 = 0.1431 ft2 0.1431 ft2 x 8.25 ft = 1.1806 ft3 1.1806 ft3 x 23.0 lb/ft3 = 27.2 lb

  47. EXERCISE 28–4 8. A 0.460-inch diameter brass rod is long. Round answers a to 3 significant digits. • Find the volume, in cubic inches, of the rod. = 0.1595 in2 0.7854 x 0.4602 0.1595 in2 x 101.21875 in = 16.144 in3

  48. EXERCISE 28–4 8. A 0.460-inch diameter brass rod is long. Round answers b to 3 significant digits. b. Compute the total weight of 40 rods. Brass weighs 0.300 pound per cubic inch. Volume = 16.144 in3 16.144 in3 x 0.300 lb/in3 = 4.843 lb each 4.843 lb x 40 rods = 194 lb

  49. Application of Volume Measure to Irregular Forms In addition to regular solids, many objects are a combination of various shapes in modified form. The volume of an irregular solid can be found by dividing it into solids having regular shapes.

  50. Application of Volume Measure to Irregular Forms The volume of each regular or modified solid form can be computed The sum of the separate volumes equals the volume of the irregular solid. The sum of the parts is equal to the whole.

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