course Mth 271 ¸èÝí´ùŒ®×à—ÆÅšÞ©ñ{™ÐÁëassignment #005
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17:15:59 explain why the slope of a depth vs. time trapezoid represents the average rate of change of the depth with respect to the time during the time interval represented
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RESPONSE --> the slope of a depth vs. time trapezoid represents the average rate of change of the depth because the graph represents the amount of depth change over a given time. dividing the depth change by the time elapsed is the same fraction as rise over run. confidence assessment: 2
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17:16:09 the specific idea is that ave rate of depth change is [change in depth / change in time] ; rise represents change in depth and run represents change in time so slope = rise/run represents ave rate of depth change. **
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RESPONSE --> self critique assessment: 3
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17:18:35 explain why the area of a rate vs. time trapezoid for a given time interval represents the change in the quantity corresponding to that time interval
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RESPONSE --> finding the area of a trapezoid involves multiplying the average altitude by the base. this is the same process as multiplying the average rate by the time elapsed. confidence assessment: 3
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17:18:55 The average altitude represents the avg. velocity. The area of a trapezoid involves the altitude, which represents the avg. velocity, and the width, which represents the change in clock time. When you multiply ave altitude by width you are representing ave vel * change in clock time, which gives change in position. This reasoning isn't confined to velocities. For any rate vs. clock time graph, average altitude represents approximate average rate, which multiplied by the change in time (not by the time itself) gives you the change in quantity **
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RESPONSE --> self critique assessment: 3
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17:26:13 text problem 0.5 #8 add x/(2-x) + 2/(x-2)
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RESPONSE --> x/(2-x) + 2/(x-2) I find the least common denominator (x^2 - 2x) / ((2-x)(x-2)) + (4 - 2x) / ((2-x)(x-2)) add the fractions (x^2 - 4x + 4) / ((2-x)(x-2)) factor the numerator (x - 2)^2 / ((2-x)(x-2)) divide (x - 2) / (2 - x) confidence assessment: 2
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17:30:52 common denominator could be [ (2-x)(x-2) ]. In this case we have x / (2-x) + 2 / (x-2) = [ (x-2) / (x-2) ] * [ x / (2-x) ] + [ (2-x) / (2-x) ] * [ 2 / (x-2) ] = x(x-2) / [ (2-x)(x-2) ] + 2 (2-x) / [ (2-x)(x-2) ] = [x(x-2) + 2(2-x) ] / [ (2-x)(x-2) ] = [ x^2 - 2x + 4 - 2x ] / [ (2-x)(x-2) ] = (x^2-4x+4) / [ -x^2+4x-4 ] = (x-2)^2 / [-(x-2)^2] = -1. NOTE however that there is a SIMPLER SOLUTION: We can note that x-2 = -(2-x) so that the original problem is -x/(x-2) + 2 /(x-2) = (-x + 2) / (x-2) = -(x-2)/(x-2) = -1. **
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RESPONSE --> My answer of (x -2) / (2 - x) was almost simplified. I could have rewritten the fraction -(2 - x) / (2 - x) and divided to get -1. self critique assessment: 2
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17:33:58 text problem 0.5 #48 cost = 6 x + 900,000 / x, write as single fraction and determine cost to store 240 units
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RESPONSE --> a) cost = 6x^2 / x + 900,000 / x cost = (6x^2 + 900,000) / x b) cost(240 units) = [(6 * 240^2) + 900,000] / 240 = $5,190 confidence assessment: 3
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17:34:13 express with common denominator x: [x / x] * 6x + 900,000 / x = 6x^2 / x + 900,000 / x = (6x^2 + 900,000) / x so cost = (6x^2+900,000)/x Evaluating at x = 240 we get cost = (6 * 240^2 + 900000) / 240 = 5190. **
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RESPONSE --> self critique assessment: 3
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