10 6 Changing Dimensions Warm Up Problem of

10 -6 Changing Dimensions Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

10 -6 Changing Dimensions Warm Up Find the surface area of each figure to the nearest tenth. Use 3. 14 for . 1. a cylinder with radius 5 ft and height 11 ft 502. 4 ft 2 2. a rectangular prism 9 ft by 14 ft by 6 ft 528 ft 2

10 -6 Changing Dimensions Problem of the Day If 6 cats can catch 6 mice in 6 minutes, how many cats are needed to catch 10 mice in 10 minutes? 6 cats

10 -6 Changing Dimensions

10 -6 Changing Dimensions Learn to find the volume and surface area of similar three-dimensional figures.

10 -6 Changing Dimensions Recall that similar figures have proportional side lengths. The surface areas of similar three-dimensional figures are also proportional. To see this relationship, you can compare the areas of corresponding faces of similar rectangular prisms.

10 -6 Changing Dimensions Area of front of smaller prism l·w 3· 5 15 Area of front of larger prism l·w 6 · 10 (3 · 2) · (5 · 2) Each dimension (3 · 5) · (2 · 2) has a scale factor of 2. 15 · 22 Remember! A scale factor is a number that every dimension of a figure is multiplied by to make a similar figure.

10 -6 Changing Dimensions The area of the front face of the larger prism is 22 times the area of the front face of the smaller prism. This is true for the entire surface area of the prisms.

10 -6 Changing Dimensions Additional Example 1 A: Finding the Surface Area of a Similar Figure The surface area of a box is 35 in 2. What is the surface area of a similar box that is larger by a scale factor of 7? S = 35 · 72 Multiply by the square of the scale factor. S = 35 · 49 Evaluate the power. S = 1, 715 Multiply. The surface area of the larger box is 1, 715 in 2.

10 -6 Changing Dimensions Additional Example 1 B: Finding the Surface Area of a Similar Figure The surface area of a box is 1, 300 in 2. Find the surface area of a similar box that is 1 smaller by a scale factor of 2. 2 Multiply by the square of the scale 1 S = 1, 300 · factor. 2 S = 1, 300 · 1 4 Evaluate the power. S = 325 Multiply. The surface area of the smaller box is 325 in 2.

10 -6 Changing Dimensions Check It Out: Example 1 A The surface area of a box is 50 in 2. What is the surface area of a similar box that is larger by a scale factor of 3? S = 50 · 32 Multiply by the square of the scale factor. S = 50 · 9 Evaluate the power. S = 450 Multiply. The surface area of the larger box is 450 in 2.

10 -6 Changing Dimensions Check It Out: Example 1 B The surface area of a box is 1, 800 in 2. Find the surface area of a similar box that is smaller 1 by a scale factor of. 3 1 2 Multiply by the square of the scale S = 1, 800 · factor. 3 S = 1, 800 · 1 9 Evaluate the power. S = 200 Multiply. The surface area of the smaller box is 200 in 2.

10 -6 Changing Dimensions The volumes of similar three-dimensional figures are also related. 2 ft 1 ft 3 ft 4 ft 2 ft 6 ft Volume of smaller box larger box 2· 3· 1 4· 6· 2 (2 · 2) · (3 · 2) · (1 · 2) 6 The volume of the larger box(2 · 3 · 1) · (2 · 2) is 23 times the volume of the 6 · 23 Each dimension smaller box. has a scale factor of 2.

10 -6 Changing Dimensions

10 -6 Changing Dimensions Additional Example 2: Finding Volume Using Similar Figures The volume of a child’s swimming pool is 28 ft 3. What is the volume of a similar pool that is larger by a scale factor of 4? V = 28 · 43 Multiply by the cube of the scale factor. V = 28 · 64 Evaluate the power. V = 1, 792 ft 3 Multiply. Estimate V ≈ 30 · 60 Round the measurements. = 1, 800 The answer is reasonable.

10 -6 Changing Dimensions Check It Out: Example 2 The volume of a small hot tube is 48 ft 3. What is the volume of a similar hot tub that is larger by a scale factor of 2? V = 48 · 23 Use the volume of the smaller prism and the cube of the scale factor. V = 48 · 8 Evaluate the power. V = 384 ft 3 Multiply. Estimate V ≈ 50 · 8 Round the measurements. = 400 The answer is reasonable.

10 -6 Changing Dimensions Additional Example 3: Problem Solving Application The sink in Kevin’s workshop measures 16 in. by 15 in. by 6 in. Another sink with a similar shape is larger by a scale factor of 2. There are 231 in 3 in 1 gallon. Estimate how many more gallons the larger sink holds.

10 -6 Changing Dimensions Additional Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. • Compare the capacities of two similar sinks, and estimate how much more water the larger sink holds. List the important information: • The smaller sink is 16 in. x 15 in. x 6 in. • The large sink is similar to the small sink by a scale factor of 2. • 231 in 3 = 1 gal

10 -6 Changing Dimensions Additional Example 3 Continued 2 Make a Plan You can write an equation that relates the volume of the large sink to the volume of the small sink. Then convert cubic inches to gallons to compare the capacities of the sinks. Volume of large sink = Volume of small sink · (a scale factor)3

10 -6 Changing Dimensions Additional Example 3 Continued 3 Solve Volume of small sink = 16 x 15 x 6 = 1, 440 in 3 Volume of large sink = 1, 440 x 23 = 11, 520 in 3 Convert each volume into gallons: 1 gal 1, 440 in 3 x ≈ 6 gallons 3 231 in 1 gal ≈ 50 gallons 3 231 in Subtract the capacities: 50 gal – 6 gal = 44 gal 11, 520 in 3 x The large sink holds about 44 gallons more than the small sink.

10 -6 Changing Dimensions Additional Example 3 Continued 4 Look Back Double the dimensions of the small sink and find the volume: 32 x 30 x 12 = 11, 520 in 3. Subtract the volumes of the two sinks: 11, 520 – 1, 440 = 10, 080 in 3. Convert this measurement to gallons: 10, 080 x 1 gal ≈ 44 gal 231 in 3

10 -6 Changing Dimensions Check It Out: Example 3 The bath tub in Ravina’s house measures 46 in. by 36 in. by 24 in. Another bath tub with a similar shape is smaller by a scale factor of 1. There are 2 231 in 3 in 1 gallon. Estimate how many more gallons the larger bath tub holds.

10 -6 Changing Dimensions Check It Out: Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. • Compare the capacities of two similar tubs, and estimate how much more water the larger tub holds. List the important information: • The larger tub is 46 in. x 36 in. x 24 in. • The smaller tub is similar to the larger tub by a scale factor of 12. • 231 in 3 = 1 gal

10 -6 Changing Dimensions Check It Out: Example 3 Continued 2 Make a Plan You can write an equation that relates the volume of the small tub to the volume of the large tub. The convert cubic inches to gallons to compare the capacities of the tubs. Volume of small tub = Volume of large tub · (a scale factor)3

10 -6 Changing Dimensions Check It Out: Example 3 Continued 3 Solve Volume of large tub = 46 x 36 x 24 = 39, 744 in 3 Volume of small tub = 39, 744 x 0. 53 = 4, 968 in 3 Convert each volume into gallons: 1 gal 39, 744 in 3 x ≈ 172 gallons 3 231 in 1 gal 4, 968 in 3 x ≈ 22 gallons 3 231 in Subtract the capacities: 172 gal – 22 gal = 150 gal The large tub holds about 150 gallons more than the small tub.

10 -6 Changing Dimensions Check It Out: Example 3 Continued 4 Look Back Halve the dimensions of the large tub and find the volume: 23 x 18 x 12 = 4, 968 in 3. Subtract the volumes of the two tubs: 39, 744 – 4, 968 = 34, 776 in 3. Convert this measurement to gallons: 34, 776 x 1 gal ≈ 150 gal. 231 in 3

10 -6 Changing Dimensions Lesson Quizzes Standard Lesson Quiz for Student Response Systems

10 -6 Changing Dimensions Lesson Quiz: Part I Given the scale factor, find the surface area to the nearest tenths of the similar prism. 1. The scale factor of the larger of two similar triangular prisms is 8. The surface area of the smaller prism is 18 ft 2. 1, 152 ft 2 2. The scale factor of the smaller of two similar 1 triangular prisms is. The surface area of the 3 larger prism is 600 ft 2. 66. 7 ft 2

10 -6 Changing Dimensions Lesson Quiz: Part II Given the scale factor, find the volume of the similar prism. 3. The scale factor of the larger of two similar rectangular prisms is 3. The volume of the smaller prism is 12 cm 3. 324 cm 3 4. A food storage container measures 6 in. by 10 in. by 2 in. A similar container is reduced by a 1 scale factor of 2. Estimate how many more gallons the larger container holds. about 0. 5 gal

10 -6 Changing Dimensions Lesson Quiz for Student Response Systems 1. A fish storage aquariumcontainer measures 15 in. by 17 in. by 7 in. A similar container aquarium is larger by a scale factor of 4. Estimate how many more gallons the larger container aquarium holds. A. about 494 gal B. about 490 gal C. about 487 gal D. about 486 gal

10 -6 Changing Dimensions Lesson Quiz for Student Response Systems 2. The volume of a prism is 28 cm 3. What is the volume of a similar prism that is larger by a scale factor 4? A. 32 cm 3 B. 112 cm 3 C. 448 cm 3 D. 1, 792 cm 3