Aim #7: What are the properties of volume of general cylinders? CC Geometry H Do Now: 1) The polygonal region has been divided into triangles. Determine the area of the entire polygon. 12 20 5 9 11 22 2) The polygon from above is used here as the base of a general right prism. Use a height of 10 and the appropriate value(s) from question 1 to determine the volume of the prism. Recall: The volume formula for a general cylinder is: (Area of the base) x (height) Find the volume to the nearest tenth. 1. 2. 8 3 10 5 6 7 3. 4. 4. 10.5 10 8.3 bases are regular hexagons
Read the comparisons between the properties of area and volume of twodimesional and three-dimensional figures. Area Properties The area of a set in two dimensions is a number greater than or equal to zero that measures the size of the set and not the shape. Volume Properties The volume of a set in three dimensions is a number greater than or equal to zero that measures the size of the set and not the shape. The area of a rectangle is given by the formula: length x width. The area of a triangle is given by the formula: ½ base x height. A polygonal region is the union of finitely many non-overlapping triangular regions and has area the sum of the areas of the triangles A right rectangular or triangular prism has volume given by the formula area of base x height. A right prism is the union of finitely many non-overlapping right rectangular or triangular prisms and has volume the sum of the volumes of the prisms Congruent regions have the same area. Congruent solids have the same volume.
Area Properties Area(A B) = Volume Properties Vol(A B) = Area shaded region = Volume surrounding A contained in solid B=
5. Paul is designing a mold for a concrete block to be used in a custom landscaping project. The block is shown in the diagram with its corresponding dimensions and consists of two intersecting rectangular prisms. Find the volume of mixed concrete, in cubic feet, needed to make Paul s custom block. 6. The base of the prism shown in the diagram consists of overlapping congruent equilateral triangles ABC and DGH. Points C, D, E and F are midpoints of the sides of triangles ABC and DGH. GH = AB = 4, and the height of the prism is 7. Find the volume of the prism.
The density of a substance is: mass volume weight 7. A square metal plate has a density of 10.2 g/cm 3 and weighs 2.193 kg. a. Change the weight to g. (1 kg = 1000 grams) b. Calculate the volume of the plate. c. If the base of this plate has an area of 25 cm 2, determine its thickness (its height). 8. Trees that are used for timber are approximately cylindrical. A timber company specializes in a certain type of tree that has a typical diameter of 50 cm and a typical height of about 10 meters. The density of the wood is 380 kilograms per cubic meter, and the wood can be sold by mass at a rate of $4.75 per kilogram. Determine and state the minimum number of whole trees that must be sold to raise at least $50,000.
Name Date 1. Find the volume of a triangle with side lengths 3, 4 and 5. CC Geometry H HW #7 2. Find the volume of the triangular prism: 3. Tim has a rectangular prism with a length of 10 cm, a width of 2 cm, and an unknown height. He needs to build another rectangular prism with a length of 5 cm and the same height as the original prism. The volume of the two prisms will be the same. Find the width, in cm, of the new prism. 4. Find the volume of a prism whose base is an equilateral triangle with a side of 4 cm and whose height is 6 cm. 5. A cylinder has a volume of 502.4 cubic feet and a diameter of 8 feet. What is the approximate height of the cylinder to the nearest tenth?
6. Two congruent solids S 1 and S 2 have the property that S 1 S 2 (their intersection) is a right triangular prism with height and a base that is an equilateral triangle of side length 2 units. If the volume of S 1 U S 2 (their union) is 25 units 3, find the volume of S 1. Mixed Review: 1) Find the value of x: 2) Find the value of x to the nearest tenth: 3) Find the perimeter to the nearest tenth: