Topic 5 Scale Factors and Volumes of 3D Objects Unit 8 Topic 5

Explore Volume is the number of cubic units inside a 3D object. Volume is measured in units Start by building each 3D object using cube links. To find the volume, determine the number of cubes required to make the object. Complete the table below. Try this on your own first!!!!

Explore 1. Complete the table below. Try this on your own first!!!!

Explore 2. The original cube has a volume of 1 unit 3. How could you calculate the volume of the other cubes using the volume of the original cube, 1 unit 3, and the linear scale factors? You should notice that the volume scale factor is equal to the linear scale factor cubed.

Information The relationship between the volume of a new 3D object and the volume of the original 3D object can be expressed using the following equation. Volume Scale Equation new volume = old volume  k 3 where k is the linear scale factor. We can rearrange the equation to isolate the volume scale factor, k 3. Volume Factor (VSF)

Example 1 Linear Scale Factor and Volume a) What is the volume of a rectangular prism with the dimension 5 cm by 4 cm by 2 cm? (V=lwh) b) The rectangular prism is enlarged by a linear scale factor of 3. What is the volume of the new prism? Method 1: Using Volume Calculation Find the volume using the dimensions of the new prism. Try this on your own first!!!!

Example 1a: Solution Linear Scale Factor and Volume What is the volume of a rectangular prism with the dimension 5 cm by 4 cm by 2 cm? (V=lwh)

Example 1b: Solution The rectangular prism is enlarged by a linear scale factor of 3. What is the volume of the new prism? Method 1: Using Volume Calculation Find the volume using the dimensions of the new prism.

Example 1b: Solution The rectangular prism is enlarged by a linear scale factor of 3. What is the volume of the new prism? Method 1: Using Volume Calculation Find the volume using the dimensions of the new prism.

Example 1b: Solution Method 2: Using the Volume Scale Equation Substitute into the volume scale equation

Example 2 Volume from a Scale Factor a)A spherical balloon is blown up. Near the start, the volume of air in the balloon is cm 3. When fully blown, the diameter of the balloon has increased by a scale factor of 3. What is the volume of the air in the balloon when fully blown? Answer to the nearest tenth of a cubic centimeter. b)The volume of ice cream inside a small cone is 40 cm3. A large cone is produced by increasing each dimension by a factor of 1.5. What is the volume of ice cream inside the large cone? Try this on your own first!!!!

Example 2a: Solution Volume from a Scale Factor A spherical balloon is blown up. Near the start, the volume of air in the balloon is cm 3. When fully blown, the diameter of the balloon has increased by a scale factor of 3. What is the volume of the air in the balloon when fully blown? Answer to the nearest tenth of a cubic centimeter.

Example 2b: Solution Volume from a Scale Factor The volume of ice cream inside a small cone is 40 cm 3. A large cone is produced by increasing each dimension by a factor of 1.5. What is the volume of ice cream inside the large cone?

Example 3 Determining Scale Factors The volume of a large rectangular bedroom is 1250 m 3. A smaller rectangular bedroom, with the same shape, has a volume 640 m 3. Determine the specified scale factor applied to the larger bedroom to produce the smaller bedroom. a)What is the linear scale factor? b)What is the surface area scale factor? c)What is the volume scale factor? Try this on your own first!!!! Helpful Hint If you know one scale factor, k, k 2 or k 3, then you can find the other two scale factors.

Example 3a: Solution Determine the specified scale factor applied to the larger bedroom to produce the smaller bedroom. a) What is the linear scale factor?

Example 3b & c: Solution b) What is the surface area scale factor? c) What is the volume scale factor? From step a) Or using the linear scale factor

Example 4 Another Example of Determining Scale Factors A small cylindrical oil storage tank has a volume of m 3. A larger cylindrical oil storage tank, with the same shape, has a volume m 3. Determine the specified scale factor applied to the smaller tank to produce the larger tank, rounded to the nearest whole number. a) What is the linear scale factor? b) What is the surface area scale factor? c) What is the volume scale factor? Try this on your own first!!!!

Example 4a: Solution Determine the specified scale factor applied to the smaller tank to produce the larger tank, rounded to the nearest whole number. a) What is the linear scale factor?

Example 4b & c: Solution b) What is the surface area scale factor? c) What is the volume scale factor? From step a) Or using the linear scale factor

Need to Know: The volume scale factor, VSF, of a 3D object is The volume of the original or old shape is multiplied by the volume scale factor to produce volume of the new shape. The volume scale equation is new volume = old volume  k 3, where k is the linear scale factor. You’re ready! Try the homework from this section.