#$&* course Mth 152 12/16/2012 6:05PM 026. ``q Query 26
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Given Solution: `a The size ratios for scale factor 2 is 4, for 3 it is 9, 4 is 16, 5 is 25, 6 is 36, 10 is 100. The size ratio is the square of the scale factor. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q **** Explain in your own words why this relationship exists. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This relationship exists because when to make a square you need the same number of squares on each side, so to find the area you find the square of the number of squares it takes to make up one side. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a In order to form a larger square by adding to the smaller one it must have the same number of edges across the top, bottom, and both sides to stay square. The size or the area of the square is found by multplying the length times the width. A square's length is the same as its width, so all you are doing is squaring one side to get the size. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Query 9.8.15 putting unit cubes together to make next larger cube **** What are the scale factor and size ratio for the two cubes? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The scale factor is 2 so the size ratio is 2 cubed, which is 8. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The scale factor is 2 and the size ratio is 8. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Query 9.8.18 dimension of cube **** What is the dimension of a cube? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The dimension of a cube is 3, because 3 is the power that the scale factor must be raised to to get the size ratio. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The dimension of a cube is the power to which the scale factor must be raised to get the size ratio. For example a cube with a scale factor of 2 would have a size ratio of 2 * 2 * 2 or 2^3. We raise the scale factor to the power 3 to get the size ratio. So the dimension is 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q **** How does the relationship between size factor and scale factor tell you that the cube is 3-dimensional? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The relationship between the size ratio and scale factor is the scale factor cubed gives us the size ratio. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a Because you have to cube the scale factor to get the size ratio. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q Query 9.8.24 Sierpinski gasket **** What are the length factor and the size factor for this figure, and what two whole numbers therefore must its dimensions therefore lie between? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a If you double the length you get 3 additional copies of the original figure. So length factor and size factor are 2 and 3. The dimension is the number such that (scale factor) ^ dimension = size factor. For example for the cube a doubling of scale factor increased size factor to 8 times its original value. This gives us the equation 2^d = 8, and as we saw above d = 3 for a cube. Here scale factor is 2 and size factor is 3 so we need to find d such that 2^d = 3. Since 2^1 = 2 and 2^2 = 4, d must be between 1 and 2. ** **** When you double the scale of the gasket by doubling its width, how many new copies of the original figure do you get? ** You get 3 copies of the original figure. ** **** Since a doubling of a scale increases size by factor 3, is the dimension greater or less than 1, and is the dimension greater or less than 2? *&*& If the dimension was 1 then doubling the scale would double the size. If the dimension was 2 then doubling the scale would give you 2^2 = 4 times the size. Since doubling gives you 3 times the size, the dimension must be greater than 1 and less than 2. *&*& **** What equation would you solve to get the dimension? ** The equation (see above note) is 2^d = 3. The solution is about d = 1.59, as you say below. ** **** Note that the equation is 2^d = 3. What approximate value of d makes this equation true? *&*& By trial and error we find that d = 1.585 comes close to making this equation true. *&*& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q Query 9.8.24 Sierpinski gasket **** What are the length factor and the size factor for this figure, and what two whole numbers therefore must its dimensions therefore lie between? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a If you double the length you get 3 additional copies of the original figure. So length factor and size factor are 2 and 3. The dimension is the number such that (scale factor) ^ dimension = size factor. For example for the cube a doubling of scale factor increased size factor to 8 times its original value. This gives us the equation 2^d = 8, and as we saw above d = 3 for a cube. Here scale factor is 2 and size factor is 3 so we need to find d such that 2^d = 3. Since 2^1 = 2 and 2^2 = 4, d must be between 1 and 2. ** **** When you double the scale of the gasket by doubling its width, how many new copies of the original figure do you get? ** You get 3 copies of the original figure. ** **** Since a doubling of a scale increases size by factor 3, is the dimension greater or less than 1, and is the dimension greater or less than 2? *&*& If the dimension was 1 then doubling the scale would double the size. If the dimension was 2 then doubling the scale would give you 2^2 = 4 times the size. Since doubling gives you 3 times the size, the dimension must be greater than 1 and less than 2. *&*& **** What equation would you solve to get the dimension? ** The equation (see above note) is 2^d = 3. The solution is about d = 1.59, as you say below. ** **** Note that the equation is 2^d = 3. What approximate value of d makes this equation true? *&*& By trial and error we find that d = 1.585 comes close to making this equation true. *&*& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: #*&!