Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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 rectangular prism 9 ft by 14 ft by 6 ft 502.4 ft2 528 ft2

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

Sunshine State Standards MA.7.G.4.1 Determine how changes in dimensions affect the…volume of common geometric figures and apply these relationships to solve problems.

Recall that similar figures have proportional side lengths 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.

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

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.

Additional Example 1: Finding the Surface Area of a Similar Figure The surface area of a box is 35 in2. What is the surface area of a 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. S = 1,715 Multiply. The surface area of the larger box is 1,715 in2.

Check It Out: Example 1A The surface area of a box is 1,800 in . Find the surface area of a similarly shaped box that is smaller by a scale factor of . 2 1 3 1 3 2 S = 1,800 1 9 S = 1,800 2 = 200 in

Check It Out: Example 1B Two rock-climbing playsets on a playground are shaped like pyramids. The surface area of one is 60 yd . What is the surface area, to the nearest tenth, of the other pyramid if it is larger by a scale factor of 2? 2 S = (60)(2) 2 S = 240 yd = (60)(4)

The volumes of similar three-dimensional figures are also related. The volume of the larger tank is 23 times the volume of the smaller tank.

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 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 ft3 Multiply. Estimate V ≈ 30 · 60 Round the measurements. = 1,800 The answer is reasonable.

Check It Out: Example 2 The volume of a small hot tub is 48 ft3 . What is the volume of a similar hot tub that is larger by a scale factor of 2? V = 48(2)3 V = 48(8) = 384 ft3

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.

Understand the Problem 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 in3 = 1 gal

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

Additional Example 3 Continued Solve 3 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: 1,440 in3 x ≈ 6 gallons 1 gal 231 in3 11,520 in3 x ≈ 50 gallons 1 gal 231 in3 Subtract the capacities: 50 gal – 6 gal = 44 gal The large sink holds about 44 gallons more than the small sink.

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 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

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 39,744 ÷ 231 ≈ 172 gal 172 - 22 = 150 V = 46(36)(24) = 39,744 in R 3

Check It Out: Example 3 Continued 1 8 V = 46(36)(24) = 4,968 in3 s 4,968 ÷ 231 ≈ 22 gal The large tub holds about 150 gallons more than the small tub.

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 22 22 22

Lesson Quiz: Part I Given the scale factor, find the surface area, to the nearest tenth, 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

Lesson Quiz: Part II 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

Lesson Quiz for Student Response Systems 1. A fish storage aquarium container 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 25 25 25

Lesson Quiz for Student Response Systems 2. The volume of a prism is 28 cm3. What is the volume of a similar prism that is larger by a scale factor 4? A. 32 cm3 B. 112 cm3 C. 448 cm3 D. 1,792 cm3 26 26 26