#$&* course Mth 163 2/18/13 around 8p.m. 011.
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Given Solution: A common response is that it takes 2 one-ft. squares to make a 2-foot square. However, below thought shows that this isn't the case. If we put 2 one foot squares side by side we get a one-foot by two-foot rectangle, not a square. If we put a second such rectangle together with the first, so that we have 2 rows with 2 squares in a row, then we have a two-foot square. Thus we see that it takes 4 squares one foot on a side to make a square 2 ft. on a side. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I understand why I missed this. That makes sense ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q002. How many cubes one foot on a side would it take to construct a cube two feet on a side? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We could begin by constructing two rows with two cubes in a row, which would sit on a square two feet by two feet. However this would not give is a cube two feet on a side, because at this point the figure we have constructed is only one foot high. So we have to add a second layer, consisting of two more rows with two cubes a row. Thus we have 2 layers, each containing 2 rows with 2 cubes in a row. Each layer has 4 cubes, so our two layers contain 8 cubes. STUDENT QUESTION The question asks how many one foot cubes to make a cube that is 2 feet on a side. It doesn’t say which side. So I don’t understand why the length of the cube can’t be 2 foot and the height only 1 foot. Was it supposed to be 2 foot on ALL sides? INSTRUCTOR RESPONSE Just as all sides of a square are the same, by definition of a square, all the sides of a cube are the same, by the definition of a cube. (Note on a technicality: technically we should call the sides of a cube 'edges'; however for present purposes the word 'side' is more familiar to most students and is used here.) &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I assumed since it was a cube that all of the sides were the same length. I do not understand how this happens or the Instructors comments on this document. ------------------------------------------------ Self-critique rating:
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Given Solution: We would require three rows, each with 3 squares, for a total of 9 squares. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q004. How many cubes one foot on a side would take to construct a cube three feet on a side? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 27 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: This would require three layers to make a cube three feet high. Each layer would have to contain 3 rows each with three cubes. Each layer would contain 9 cubes, so the three-layer construction would contain 27 cubes. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q005. Suppose one of the Egyptian pyramids had been constructed of cubical stones. Suppose also that this pyramid had a weight of 100 million tons. If a larger pyramid was built as an exact replica, using cubical stones made of the same material but having twice the dimensions of those used in the original pyramid, then what would be the weight of the larger pyramid? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 200 million tons confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Each stone of the larger pyramid has double the dimensions of each stone of the smaller pyramid. Since it takes 8 smaller cubes to construct a cube with twice the dimensions, each stone of the larger pyramid is equivalent to eight stones of the smaller. Thus the larger pyramid has 8 times the weight of the smaller. Its weight is therefore 8 * 100 million tons = 800 million tons. STUDENT QUESTION I totally missed the mark on this one. Where did the 8 cubes come from (how is that calculated?)? I think that’s what is throwing me off. INSTRUCTOR RESPONSE If each pyramid is divided into the same number of cubes, in a geometrically similar manner, then the cubes of the second pyramid will have double the dimensions of the cubes making up the first. If you double the dimensions of a cube, the larger cube could be built from 2 * 2 * 2 = 8 of the smaller, as you saw on a previous question. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I understand why I missed this and realize what I did wrong ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q006. Suppose that we wished to paint the outsides of the two pyramids described in the preceding problem. How many times as much paint would it take to paint the larger pyramid? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 8 times as much as the first pyramid confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The outside of each pyramid consists of square faces of uniform cubes. Since the cubes of the second pyramid have twice the dimension of the first, their square faces have 4 times the area of the cubes that make up the first. There is therefore 4 times the area to paint, and the second cube would require 4 times the paint STUDENT COMMENT: I’m not getting this concept at all. INSTRUCTOR RESPONSE: Normally I ask for more detail in a self-critique so I can tell what you do and do not understand, and give you the answer you need to see. However in this case you have been doing well to this point, and self-critiquing when necessary, so I have a basis on which to focus my answer: If the surface of each pyramid is divided into the same number of squares, in a geometrically similar manner, then the squares of the second pyramid will have double the dimensions of the squares making up the first. If you double the dimensions of a square, the larger square could be built from 2 * 2 = 4 of the smaller. STUDENT COMMENT I don’t get it. Why is the one problem 2 * 2 * 2 and this one only 2 * 2. I don’t understand why the difference. INSTRUCTOR RESPONSE Surfaces can be covered as by tiny squares. Volumes can be filled by tiny cubes. Paint covers a surface, it doesn't fill the volume. So we have to think in terms of covering the surface tiny squares, not filling the volume with tiny cubes. Imagine a picture of a surface covered by tiny squares. If we double the dimensions of the picture, the sides of the tiny squares double. Doubling the sides of a square gives you 4 times the area. Thus when we double the dimensions, the area becomes 4 times as great. (Technicality: We can't really cover a surface with curved boundaries using squares. At least near the curved boundaries, the squares will always make inconvenient angles with one another, and will therefore either overlap or leave small gaps. However the smaller we make the squares the smaller the total area of the overlaps or gaps, and it turns out we can cover the surface leaving as little total overlap or gap as we wish, and as we approach the limit of 'tiny' the total area the the squares approaches the actual area of the surface as its limit.) &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I was thinking more of the cube with volume instead of the paint being just the surface. I understand now ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Suppose that we know that y = k x^2 and that y = 12 when x = 2. What is the value of k? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To obtain the value of k we must plug in the values we know into the equation getting 12 = k*2^2. We square 2 and divide 12 by the answer getting k = 3. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To find the value of k we substitute y = 12 and x = 2 into the form y = k x^2. We obtain 12 = k * 2^2, which we simplify to give us 12 = 4 * k. The dividing both sides by 410 reversing the sides we easily obtain k = 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 2 ********************************************* Question: `q008. Substitute the value of k you obtained in the last problem into the form y = k x^2. What equation do you get relating x and y? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If we substitute the value of k in we get y = 3*x^2. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: We obtained k = 3. Substituting this into the form y = k x^2 we have the equation y = 3 x^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q009. Using the equation y = 3 x^2, determine the value of y if it is known that x = 5. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Using that equation we just plug in 5 for x getting the equation y = 3*5^2. We squre 5 getting 25 then multiply by 3 getting y = 75 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If x = 5, then the equation y = 3 x^2 give us y = 3 (5)^2 = 3 * 25 = 75. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q010. If it is known that y = k x^3 and that when x = 4, y = 256, then what value of y will correspond to x = 9? To determine your answer, first determine the value of k and substitute this value into y = k x^3 to obtain an equation for y in terms of x. Then substitute the new value of x. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We can use the know values of x = 4 and y = 256 to find what k equals. When we do this we plug in the values getting the equation 256 = k*4^3. Solving for k we get k = 4. Then we can plug in 4 into the equation obtained giving the equation y = 4 *9^3. When we do this we cube 9 and multiply by 4. This gives us y = 2,916. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To we first substitute x = 4, y = 256 into the form y = k x^3. We obtain the equation 256 = k * 4^3, or 256 = 64 k. Dividing both sides by 64 we obtain k = 256 / 64 = 4. Substituting k = 4 into the form y = k x^3, we obtain the equation y = 4 x^3. We wish to find the value of y when x = 9. We easily do so by substituting x equal space 9 into our new equation. Our result is y = 4 * 9^3 = 4 * 729 = 2916. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q011. If it is known that y = k x^-2 and that when x = 5, y = 250, then what value of y will correspond to x = 12? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In order to determine this we must again solve for k. When we do this we have the equation 250 = k*5^-2. When we solve for k we get k = 6250. Plugging this value in to the new equation of y = 6250 * 12^-2. When we solve for y we get y = 43.4 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Substituting x = 5 and y = 250 into the form y = k x^-2 we obtain 250 = k * 5^-2. Since 5^-2 = 1 / 5^2 = 1/25, this becomes 250 = 1/25 * k, so that k = 250 * 25 = 6250. Thus our form y = k x^-2 becomes y = 6250 x^-2. When x = 12, we therefore have y = 6250 * 12^-2 = 6250 / 12^2 = 6250 / 144 = 42.6, approximately. STUDENT QUESTION When I enter 6250/144 in my calculator, I get 43.40277 I don’t understand where the solution 42.6 came from. Shouldn’t it be 43.4? INSTRUCTOR RESPONSE The key word in the given solution is 'approximately'. My numbers are typically accurate to within a few percent, but are often off by as much as that. This is intentional, as it shows me when students don't do their own calculations. My numbers are therefore guidelines. They are often accurate, and when not precise they should be accurate enough to show you when you are doing something incorrectly. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q012. A sculpture is made of small cubes, about the size of playing dice. Suppose each small cube could be cut into three smaller cubes, with each of the smaller cubes then expanding in size until it is the size of the original to form a larger sculpture. The shape of the sculpture would not change, only its size would be different. Don't worry about how this might be accomplished or even whether this is possible. If this could be done, how many times higher would the larger sculpture be? How many times as much volume would the larger sculpture occupy? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It would be 3 times larger The volume would be 6 times as much