Mathematically Similar Created by Bernard Lafferty BscHons Mathematics

Mathematically Similar Created by Bernard Lafferty Bsc(Hons) Mathematics GIMA 11/24/2020 www. mathsrevision. com 1

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 the scaling factor is related to the length, area and volume of the object in different dimensions. 11/24/2020 www. mathsrevision. com 2

Mathematically Similar Consider the simple case of one dimension. Draw a line 2 unit long and then draw another 4 units long. x direction 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. 11/24/2020 www. mathsrevision. com 3

Mathematically Similar Consider the case in two dimensions. Draw a square with sides 2 units long and then draw another 4 units long. y Area = 2 x 2 = 4 x Sides 2 units long y Area = 4 x 4 = 16 x 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 k 2. In our example we have this case 22 = 4. 11/24/2020 www. mathsrevision. com 4

Mathematically Similar Consider the case in three dimensions. Draw a cube with sides 2 units long and then draw another 4 units long. y z x Volume = 2 x 2 x 2 = 8 Sides 2 units long y z Volume = 4 x 4 x 4 = 64 x 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 k 3. In our example we have this case 23 = 4. 11/24/2020 www. mathsrevision. com 5

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 k 2 Three dimensions : Volume is changed by a factor of k 3 Note : If k < 1 then the scaling is reduced in size. If k > 1 then the scaling is increased in size. 11/24/2020 www. mathsrevision. com 6

Example Q 1. An A 4 sheet of paper has area 600 mmm 2. 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 11/24/2020 A=(1/2)2 x 600 = (1/4) x 600 = 150 mm 2 www. mathsrevision. com 7

Example Q 2. An small cereal box contains 100 g 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 = 800 g. THE END 11/24/2020 www. mathsrevision. com 8