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10-5

10-5. Changing Dimensions. Course 2. Warm Up. Problem of the Day. Lesson Presentation. 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 2. a sphere with diameter 6 ft

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10-5

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  1. 10-5 Changing Dimensions Course 2 Warm Up Problem of the Day Lesson Presentation

  2. 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 2. a sphere with diameter 6 ft 3. a rectangular prism 9 ft by 14 ft by 6 ft 502.4 ft2 113.0 ft2 528 ft2

  3. 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

  4. Learn to find the volume and surface area of similar three-dimensional figures.

  5. Recall that similar figures are proportional. 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.

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

  7. 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 all of the corresponding faces. Thus it is also true for the entire surface area of the prisms. surface area of figure A surface area of figure B (scale factor)2 = •

  8. Additional Example 1A: Finding the Surface Area of a Similar Figure The surface area of a box is 35 in2. What is the surface area of a larger, similar box that is larger by a scale factor of 7? Multiply by the square of the scale factor. S = 35 · 72 S = 35 · 49 Evaluate the power. Multiply. S = 1,715 The surface area of the larger box is 1,715 in2.

  9. Additional Example 1B: Finding the Surface Area of a Similar Figure The surface area of a box is 1,300 in2. Find the surface area of a smaller, similar box that is smaller by a scale factor of . 1 2 Multiply by the square of the scale factor. 1 2 2 S = 1,300 · 1 4 S = 1,300 · Evaluate the power. S = 325 Multiply. The surface area of the smaller box is 325 in2.

  10. Check It Out: Example 1A The surface area of a box is 50 in2. What is the surface area of a larger, similar box that is larger by a scale factor of 3? Multiply by the square of the scale factor. S = 50 · 32 Evaluate the power. S =50 · 9 Multiply. S = 450 The surface area of the larger box is 450 in2.

  11. Check It Out: Example 1B The surface area of a box is 1,800 in2. Find the surface area of a smaller, similarly shaped box that has a scale factor of . 1 3 Multiply by the square of the scale factor. 1 3 2 S = 1,800 · 1 9 S = 1,800 · Evaluate the power. S = 200 Multiply. The surface area of the smaller box is 200 in2.

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

  13. volume of figure B volume of figure A (scale factor)3 • =

  14. Additional Example 2: Finding Volume Using Similar Figures The volume of a child’s swimming pool is 28 ft3. What is the volume of a similar pool prism that is larger by a scale factor of 4? Multiply by the cube of the scale factor. V = 28 · 43 Evaluate the power. V = 28 · 64 V = 1,792 ft3 Multiply. Estimate Round the measurements. V ≈ 30 · 60 = 1,800 The answer is reasonable.

  15. Check It Out: Example 2 The volume of a small hot tube is 48 ft3. What is the volume of a similar hot tub that is larger by a scale factor of 2? Use the volume of the smaller prism and the cube of the scale factor. V = 48 · 23 Evaluate the power. V = 48 · 8 V = 384 ft3 Multiply. Estimate Round the measurements. V ≈ 50 · 8 = 400 The answer is reasonable.

  16. 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 in3 in 1 gallon. Estimate how many more gallons the larger sink holds.

  17. 1 Understand the Problem Additional Example 3 Continued 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 larger sink is similar to the small sink by a scale factor of 2. •231 in3 = 1 gal

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

  19. 3 Solve 1 gal 231 in3 1 gal 231 in3 1,440 in3 x ≈ 6 gallons 11,520 in3 x ≈ 50 gallons Additional Example 3 Continued Volume of small sink = 16 x 15 x 6 = 1,440 in3 Volume of large sink = 1,440 x 23 = 11,520 in3 Convert each volume into gallons: Subtract the capacities: 50 gal – 6 gal = 44 gal The large sink holds about 44 gallons more than the small sink.

  20. 4 Additional Example 3 Continued Look Back Double the dimensions of the small sink and find the volume: 32 x 30 x 12 = 11,520 in3. Subtract the volumes of the two sinks: 11,520 – 1,440 = 10,080 in3. Convert this measurement to gallons: 10,080 x ≈ 44 gal. 1 gal 231 in3

  21. 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 . There are 231 in3 in 1 gallon. Estimate how many more gallons the larger bath tub holds. 1 2

  22. 1 Understand the Problem •The smaller tub is similar to the larger tub by a scale factor of . 1 2 Check It Out: Example 3 Continued 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. •231 in3 = 1 gal

  23. Make a Plan 2 Check It Out: Example 3 Continued 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

  24. 3 Solve 1 gal 231 in3 39,744 in3 x ≈ 172 gallons 1 gal 231 in3 4,968 in3 x ≈ 22 gallons Check It Out: Example 3 Continued Volume of large tub = 46 x 36 x 24 = 39,744 in3 Volume of small tub = 39,744 x 0.53 = 4,968 in3 Convert each volume into gallons: Subtract the capacities: 172 gal – 22 gal = 150 gal The large tub holds about 150 gallons more than the small tub.

  25. 4 Check It Out: Example 3 Continued Look Back Half the dimensions of the large tub and find the volume: 23 x 18 x 12 = 4,968 in3. Subtract the volumes of the two tubs: 39,744 – 4,968 = 34,776 in3. Convert this measurement to gallons: 34,776 x ≈ 150 gal. 1 gal 231 in3

  26. 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 ft2. 2. The scale factor of the smaller of two similar triangular prisms is . The surface area of the larger prism is 600 ft2. 1,152 ft2 1 3 66.7 ft2

  27. 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 cm3. 4. A food storage container measures 6 in. by 10 in. by 2 in. A similar container is reduced by a scale factor of . Estimate how many more gallons the larger container holds. 324 cm3 1 2 About 0.5 gal

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