course Mth 163
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17:46:52 `q001. Note that this assignment has 11 questions How many squares one foot on a side would it take to construct a square two feet on a side?
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RESPONSE --> In order to construct a square two feet on a side from a one foot square would require 7 more squares one foot each. This would double the one foot square to make a 2 foot square or a total of 8 one foot squares.
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19:33:17 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.
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RESPONSE --> This answer is correct and I guess if I had read correctly I would have answered the same. My thought was to have a square two feet on each side. Therefore, the answer I gave listed the dimensions to complete this ""square"".
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19:35:25 `q002. How many cubes one foot on a side would it take to construct a cube two feet on a side?
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RESPONSE --> It would take 8 cubes one foot on a side to construct a cube two feet on a side. 3 more cubes on the to complete the ""bottom"" layer added to the original cube and then four to complete the upper ""level"" to have a cube 2 feet on every side.
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19:35:42 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.
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RESPONSE --> OK
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19:39:41 `q003. How many squares one foot on a side would it take to construct a square three feet on a side?
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RESPONSE --> It would take a total of 8 squares on foot on a side to construct a square three feet on a side.
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19:40:23 We would require three rows, each with 3 squares, for a total of 9 squares.
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RESPONSE --> Correct...9 total squares...8 more if the original is already included.
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19:40:56 `q004. How many cubes one foot on a side would take to construct a cube three feet on a side?
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RESPONSE --> 27 total cubes, one foot on a side to construct a cube three feet on a side.
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19:41:05 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.
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RESPONSE --> OK
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21:17:57 `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?
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RESPONSE --> The weight of the larger pyramid would be 400 million tons. In order to make the stone 2 times as big, there would have to be four of the original stones to make the size of the new stone that is 2 times as big. Multiplying 100 million tons by 4 would make 4 million tons.
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21:18:58 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.
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RESPONSE --> Correct. I was thinking of squares and not cubes. But in order to make a cube twice as big, you would have to multiply by 8 which would make the weight of the new pyramid 8 million tons.
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21:20:13 `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?
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RESPONSE --> It would take 4 times the amount of paint since only one side of the stone would be painted.
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21:20:22 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
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RESPONSE --> OK
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21:23:09 `q007. Suppose that we know that y = k x^2 and that y = 12 when x = 2. What is the value of k?
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RESPONSE --> y = k x^2 when y = 12 and x = 2. 12 = k 2^2 12 = k 4 3 = k
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21:24:06 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.
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RESPONSE --> I don't quite understand why you divide both sides by 410 unless it is just a ""typo"".
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21:26:19 `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?
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RESPONSE --> y = 3 x^2 and y / 3 = x^2
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21:27:12 We obtained k = 3. Substituting this into the form y = k x^2 we have the equation y = 3 x^2.
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RESPONSE --> The second equation was an attempt to put the ""x"" variable on its own side of the equation. By looking at your answer, I see that I ""thought"" too much.
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21:28:35 `q009. Using the equation y = 3 x^2, determine the value of y if it is known that x = 5.
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RESPONSE --> y = 3(5)^2 y = 3 (25) y = 75
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21:28:42 If x = 5, then the equation y = 3 x^2 give us y = 3 (5)^2 = 3 * 25 = 75.
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RESPONSE --> OK
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21:32:31 `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.
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RESPONSE --> 256 = k 64 4 = k y = 4 (9)^3 y = 2916
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21:32:37 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.
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RESPONSE --> OK
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21:41:02 `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?
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RESPONSE --> y = k x^-2 250 = k / 25 6250 = k y = 6250 / 144 y = 43.4028
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21:41:36 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.
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RESPONSE --> OK
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