10-6 Changing Dimensions. IWBAT find the volume and surface area of similar three-dimensional figures.
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1 IWBAT find the volume and surface area of similar three-dimensional figures.
2 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.
3 Area of front of smaller prism Area of front of larger prism l w l w (3 2) (5 2) Each dimension (3 5) (2 2) has a scale Remember! factor of 2. A scale factor is a number that every dimension of a figure is multiplied by to make a similar figure.
4 The area of the front face of the larger prism is 2 2 times the area of the front face of the smaller prism. This is true for the entire surface area of the prisms. WRITE:
5 EX 1A: 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 = S = S = 1,715 Multiply by the square of the scale factor. Evaluate the power. Multiply. The surface area of the larger box is 1,715 in 2.
6 EX 1B: 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. S = 1, Multiply by the square of the scale 2 factor. S = 1, S = 325 Evaluate the power. Multiply. The surface area of the smaller box is 325 in 2.
7 The volumes of similar three-dimensional figures are also related. 3 ft 2 ft 1 ft Volume of smaller box ft The volume of the larger box is 2 3 times the volume of the smaller box. 2 ft 6 ft Volume of larger box (2 2) (3 2) (1 2) ( 2 2 (2 1) 3 ( Each dimension has a scale factor of 2.
8 WRITE:
9 EX 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 = V = V = 1,792 ft 3 Multiply by the cube of the scale factor. Evaluate the power. Multiply. Estimate V Round the measurements. = 1,800 The answer is reasonable.
10 EX 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.
11 EX 3 Continued 1 Study the Problem and Organize the Facts 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 in 3 = 1 gal
12 2 Line Up a Plan EX 3 Continued 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
13 EX 3 Continued 3 Verify Your Plan with Action Volume of small sink = 16 x 15 x 6 = 1,440 in 3 Volume of large sink = 1,440 x 2 3 = 11,520 in 3 Convert each volume into gallons: 1,440 in 3 1 gal x 6 gallons 231 in 3 11,520 in 3 1 gal x 50 gallons 231 in 3 Subtract the capacities: 50 gal 6 gal = 44 gal The large sink holds about 44 gallons more than the small sink.
14 EX 3 Continued 4 Evaluate Your Solution 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
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Table 1. Densities of common substances in g/ml Mercury 13.6 Alcohol Lead 11.4 Ice Aluminum 2.70 Air
Worksheet I.4: Density Calculations Name: Date: Table 1. Densities of common substances in g/ml Mercury 13.6 Alcohol 0.789 Lead 11.4 Ice 0.917 Aluminum 2.70 Air 0.00120 Use the information above to answer