The volume of concrete needed to construct the staircase is 0.24 m 3. The volume of the staircase is _ 6 9 or _ 2 the volume of the square-based prism. The staircase consists of six of the nine congruent rows of cubes shown.
Think of the staircase as part of a larger square-based prism. The volume of concrete needed to construct the staircase is approximately 0.24 m Volume MHR 21 01_FFCM12_CH1.indd 21 3/6/09 12:17:24 PMĥ Method 2: Use the volume of a simpler object. 0.9 m 1350 cm 2 Volume = Base area Depth = Convert the base area to square metres: 1350 cm 2 ( ) = m 2. Multiply the base area by the depth to determine the volume of the prism. The base consists of six congruent squares = 1350 The total base area is 1350 cm 2. Determine the area of the base and then multiply by the depth. 0.9 m 45 cm Solution This staircase is in the shape of a prism with a staircase-shaped base. h = 2 ft 20 MHR Chapter 1 01_FFCM12_CH1.indd 20 3/6/09 12:17:23 PMĤ Example 3 Volume of a Composite Figure Determine the volume, to two decimal places, of concrete needed to construct this staircase. d = 2r = 2(0.885) = 1.77 The minimum diameter of the container is 1.77 m or 177 cm. V = πr 2 h 1.5 = πr 2 (0.6096) 1.5 = πr 2 _ π = _ πr2 Divide both sides by π π r = r 2 Take the square root of both sides r Method 2: Rearrange, then substitute. Substitute h = and V = 1.5 into the formula for volume of a cylinder and solve for the radius. What is the minimum diameter, to the nearest centimetre? Solution Convert the height to metres. The container is be 2 ft high, and must hold 1.5 m 3 of soil. 1.2 Volume MHR 19 01_FFCM12_CH1.indd 19 3/6/09 12:17:23 PMģ Example 2 Volume of a Cylinder A cylindrical container is to be planted with flowers. 1.2 m 45.0 cm m 2 Volume = Base area Depth = Approximately 0.7 m 3 of concrete is needed to construct the ramp. Multiply this value by the depth of the prism to determine its volume. Determine the area of the triangular base cm 2.5 m A = _ 1 (b h) 2 = _ 1 ( ) 45 cm = 0.45 m 2 = The base area of the prism is m 2. Solution The ramp is in the shape of a triangular-based prism. Depending on the orientation of the prism, the bases can be on the top and bottom, the front and back, or the left and right sides.
1.2 m 2.5 m 45.0 cm Literacy Connect The base of a prism refers to one of the two congruent, parallel polygon sides. Which structures are made up of two or more different three-dimensional figures? What threedimensional figures are used to make up each structure? 18 MHR Chapter 1 01_FFCM12_CH1.indd 18 3/6/09 12:17:22 PMĢ Example 1 Volume of a Prism Determine the volume of concrete needed to construct the ramp shown, to the nearest tenth of a cubic metre. Provide an image of each structure and indicate what geometric shapes are used. What geometric shapes can you identify? What problems could arise if incorrect measurements are used in constructing these shapes? sphere cylinder rectangular prism triangular prism Research the Internet for at least three cities or places with buildings or structures that have geometric shapes. 1 1.2 Volume When designing a building, it is important for the architect to measure carefully so that the structure functions properly and is pleasing to the eye.