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Mathematically Similar

Mathematically Similar. Created by Bernard Lafferty Bsc(Hons) Mathematics GIMA. Mathematically Similar. An object is said to be mathematically similar to another. if the only difference between them is a scaling (k) in. ALL directions.

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Mathematically Similar

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  1. Mathematically Similar Created by Bernard Lafferty Bsc(Hons) Mathematics GIMA www.mathsrevision.com

  2. Mathematically Similar An object is said to be mathematically similar to another if the only difference between them is a scaling (k) in ALL directions The objects can be complex in shape and be in one, two or three dimensions. I will use a square shape to show how the scaling factor is related to the length, area and volume of the object in different dimensions. www.mathsrevision.com

  3. x direction Mathematically Similar Consider the simple case of one dimension. Draw a line 2 unit long and then draw another 4 units long. 2 units long 4 units long It should be quite clear that second line is twice the first. Hence the scaling factor is k = 2. www.mathsrevision.com

  4. y y x x Mathematically Similar Consider the case in two dimensions. Draw a square with sides 2 units long and then draw another 4 units long. Area = 2x2 = 4 Sides 2 units long Area = 4x4 = 16 Sides 4 units long It should be quite clear that second area is four times the first. Hence the scaling factor for two dimension is k2. In our example we have this case 22 = 4. www.mathsrevision.com

  5. y y z z x x Mathematically Similar Consider the case in three dimensions. Draw a cube with sides 2 units long and then draw another 4 units long. Volume = 2x2x2 = 8 Sides 2 units long Volume = 4x4x4 = 64 Sides 4 units long It should be quite clear that Volume area is eight times the first. Hence the scaling factor for two dimension is k3. In our example we have this case 23 = 4. www.mathsrevision.com

  6. Summary If we scale a simple or complex object by a factor of k units it has the effect of the following :- YOU NEED TO REMEMBER THE FOLLOWING One dimension : Length is changed by a factor of k Two dimensions : Area is changed by a factor of k2 Three dimensions : Volume is changed by a factor of k3 Note : If k < 1 then the scaling is reduced in size. If k > 1 then the scaling is increased in size. www.mathsrevision.com

  7. Example Q1. An A4 sheet of paper has area 600mmm2. If it is cut in half both long and short ways what is the value of area left. Solution Scaling factor is k = 1/2 New area is A=(1/2)2 x 600 = (1/4) x 600 = 150mm2 www.mathsrevision.com

  8. Example Q2. An small cereal box contains 100g of cereal. How much does a large box contain if it is double the size. Assume the objects are mathematically similar. Solution Scaling factor is k = 2 New Volume is V=(2)3 x 100 = (8) x 100 = 800g. THE END www.mathsrevision.com

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