Do Now: For the following pair of similar figures, write the ratio of side lengths
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3 CC Geometry H Aim #8: What is the relationship between the ratio of the side lengths of similar solids and the ratio of their volumes? Do Now: For the following pair of similar figures, write the ratio of side lengths c:d that compares one pair of corresponding sides. Then write the ratio that compares the areas of the similar figures. Simplify ratios. When the ratio of side lengths is c : d, then the ratio of the areas is. Make a conjecture as to how the ratio of sides c : d will be related to the ratio of volumes Volume (C) : Volume (D). 1) Each pair of solids shown below is similar. Write the ratio of side lengths a:b comparing one pair of corresponding sides. Then, complete the third column by writing the ratio that compares volumes of the similar figures. Simplify ratios when possible. Similar Figures Ratio of Side Lengths Ratio of Volumes a : b Volume (A) : Volume (B) Volume of a general cone: (1/3)Area of base x height
4 Similar Figures Ratio of Side Lengths Ratio of Volumes a : b Volume (A) : Volume (B) ) b. Suppose a similarity transformation takes a solid S to a solid T at scale factor r. How do you think the volume of S compares to the volume of T? 2) Coffee sold at a deli comes in similar shaped cups. A small cup has a height of 4.2" and a large cup has a height of 5". The large coffee holds 12 fluid oz. How much coffee is in a small cup? Round your answer to the nearest tenth. 3) Right circular cylinder A has volume 2,700 and radius 3. Right circular cylinder B is similar to cylinder A and has volume 6,400. Find the radius of cylinder B.
5 4) a. Calculate the volume of this triangular prism: b. If one side of the triangular base is scaled by a factor of 2, the other side by a factor of 4, and the height of the prism by a factor of 3, what are the dimensions of the scaled prism? c. Calculate the volume of the scaled triangular prism. d. Make a conjecture about the relationship between the volume of the original prism and the scaled prism. Do the volumes have the same relationship we found in problem 1? 5) a. Calculate the volume of this rectangular prism: b. If one side of the rectangular base is scaled by a factor of 1/2, the other side by a factor of 24, and the height of the prism by a factor of 1/3, what are the dimensions of the scaled prism? c. Calculate the volume of the scaled rectangular prism. d. Make a conjecture about the relationship between the volume of the original prism and the scaled prism. e. If a solid T is scaled by factors r, s, and t in three perpendicular directions, what happens to the volume?
6 Let's Sum It up! -If the ratio of the lengths of similar solids is a : b, then the ratio of their volumes is a 3 : b 3. Volume (B) = r 3 x Volume (A) -When a solid with volume V is scaled by factors r, s, and t in three perpendicular directions, then the volume of the scaled figure is multiplied by rst. Volume (T') = rst x Volume (T)
7 Name Date 1. Two circular cylinders are similar. The ratio of the areas of their bases is 9:4. Find the ratio of the volumes of the similar solids. CC Geometry H HW #8 2. The volume of a rectangular pyramid is 60. The width of the base is then scaled by a factor of 3, the length of the base is scaled by a factor of 5/2, and the height of the pyramid is scaled such that the resulting image has the same volume as the original pyramid. Find the scale factor used for the height. 3. The following solids are similar. The volume of the 1st solid is Find the volume of the second. 4. If the sides of two similar triangles are in the ratio of 2:3, find the ratio of their perimeters and areas. 5. The ratio of the sides of two similar cubes is 3:4. the smaller cube has a volume of 729 m 3. What is the volume of the larger cube?
8 6. The Empire State Building is a 102 story skyscraper. Its height is 1,250 ft from the ground to the roof. The length and width of the building are approximately 424 ft and 187 ft, respectively. A company plans to make a miniature version of the building and sell cases to souvenir shops. a. The miniature version is just 1/2500 of the size of the original. What are the dimensions of the miniature Empire State Building? Round to the nearest hundredth when necessary. b. Determine the volume of the miniature building. Let us assume that the building is in the shape of a rectangular prism. Review: 1. For each pair of similar figures, write the ratio of the side lengths a:b. Then write the ratio that compares the areas of the similar figures. Simplify ratios. a. b. 2. From the top of an apartment building, the angle of depression to a car parked on the street below is 38 o. The car is parked 80 feet from the base of the building. Find the height of the building, to the nearest tenth of a foot.
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