course MTH 271 09.29, 1500 assignment #006006. `query 6
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23:59:12 Query class notes #06 If x is the height of a sandpile and y the volume, what proportionality governs geometrically similar sandpiles? Why should this be the proportionality?
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RESPONSE --> If sandpiles are geometrically similar, the height (x) is proportional to the volume (y). As the height increases, the volume also increases. This creates a smooth curve that is increasing at an increasing rate. confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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00:00:49 If x is the radius of a spherical balloon and y the surface area, what proportionality governs the relationship between y and x? Why should this be the proportionality?
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RESPONSE --> The relationship here is governed by the proportionality y=kx^2. confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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00:07:52 Explain how you would use the concept of the differential to find the volume of a sandpile of height 5.01 given the volume of a geometrically similar sandpile of height 5, and given the value of k in the y = k x^3 proportionality between height and volume.
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RESPONSE --> Having the x and k value for a height of 5 would allow us to calculate the y value. We can then plot this as a point on our graph. Drawing a line that is tangent to this point on the curve and determining its slope would allow us to determine the rate of change. Since there is very little variance in an x value of 5 and an x value of 5.01, we base our calculations on the assumption that the rate at these two heights is very close. Multiplying the rate of change (i.e., the slope of the tangent line...change in volume/change in height) by the change in height between 5 and 5.01, we are able to approximate a volume (y) for 5.01. confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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00:08:51 What would be the rate of depth change for the depth function y = .02 t^2 - 3 t + 6 at t = 30? (instant response not required)
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RESPONSE --> y = .02t^2 - 3t + 6 y'=.04t -3 y'(30)=.04(30) - 3 y'(30)=-1.8 cm/sec confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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00:09:30 modeling project 3 problem a single quarter-cup of sand makes a cube 1.5 inches on a side. How many quarter-cups would be required to make a cube with twice the scale, 3 inches on a side? Explain how you know this.
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RESPONSE --> If we construct a cube with 3 inch sides, we would need 8 1.5 inch cubes to accomplish this. Therefore, since a 1.5 inch cube is filled with 1 quarter-cup of sand and 8 1.5 inch cubes is need to make one 3 inch cube, 8 quarter-cups of sand would be needed. confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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00:10:06 What value of the parameter a would model this situation? How many quarter-cups does this model predict for a cube three inches on a side? How does this compare with your previous answer?
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RESPONSE --> y=ax^3 1=a (1.5^3) .296=a y=.296x^3 y=.296(3^3) y=7.992 confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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00:10:34 What would be the side measurement of a cube designed to hold 30 quarter-cups of sand? What equation did you solve to get this?
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RESPONSE --> y=.296x^3 30=.296x^3 x=4.66"" confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ query problem 2. Someone used 1/2 cup instead of 1/4 cup. The best-fit function was y = .002 x^3. What function would have been obtained using 1/4 cup?
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RESPONSE --> 1/2 cup = 2(1/4 cup) y(1/4)=.004 x^3 confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:01:54 query problem 4. number of swings vs. length data. Which function fits best?
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RESPONSE --> y=ax^-.5 best fits the data. confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:04:06 If the time per swing in seconds is y, then what expression represents the number of swings per minute?
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RESPONSE --> f=60 / y confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:06:05 If the time per swing is a x ^ .5, for the value determined previously for the parameter a, then what expression represents the number of swings per minute? How does this expression compare with the function you obtained for the number of swings per minute vs. length?
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RESPONSE --> 60 / (a x^-.5) =y confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:14:12 query problem 9. length ratio x2 / x1.
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RESPONSE --> y=a(2x^.5) ...2x length y=ax^.5...1x length confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:15:28 What expressions, in terms of x1 and x2, represent the frequencies (i.e., number of swings per minute) of the two pendulums?
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RESPONSE --> F1=55 (x1)^-.5 F2=55 (x2)^-.5 confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:19:35 What expression, in terms of x1 and x2, represents the ratio of the frequencies of the two pendulums?
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RESPONSE --> ((55 x2)^-.5) / ((55 x1)^-.5 (x2)^-.5 / (x1)^ -.5 (x1)^.5 / (x2)^.5 confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:21:15 query problem Challenge Problem for Calculus-Bound Students: how much would the frequency change between lengths of 2.4 and 2.6 feet
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RESPONSE --> f= 55 (2.4)^-.5 f=35.5 f=55 (2.6)^-.5 f=34.11 35.5 - 34.11=1.39 confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:28:06 1.1.20 (was 1.1.18 are (0,4), (7,-6) and (-5,11) collinear?
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RESPONSE --> The points are collinear. d1=(0,4) to (7,-6) =sqrt((7-0)^2 + (-6-4)^2) =sqrt(49 + 100) =sqrt (149) d1=12.2 d2=(0,4) to (-5,11) =sqrt((-5-0)^2 + (11 - 4)^2) =sqrt (25 + 49) =sqrt (74) =8.6 d3= (-5,11) to (7, -6) =sqrt ((7--5)^2 + (-6 -11)^2) =sqrt (144 + 289) =sqrt (433) =20.8 d1 + d2=d3 12.2 + 8.6 = 20.8 confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:35:17 1.1.24 find x | dist (2,-1) to (x,2) is 5What value of x makes the distance 5?
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RESPONSE --> 5=sqrt((x-2)^2 + (2- -1)^2) 5=sqrt((x^2 - 4x +4) + 9) 5=sqrt((x^2 -4x + 13)) 25=x^2 -4x + 13 0=x^2 -4x - 12 0=(x - 6) (x + 2) 0=x-6 x=6 0=x + 2 x=-2 confidence rating: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:38:01 1.1.36 (was 1.1.34 percent increase in Dow What are the requested percent increases?
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RESPONSE --> A. March '05-->Nov. '05 [(10400-10800)/10800] * 100=-3.7% B. May '06-->Feb. '07 [(12600 - 11400)/11400] * 100=+12.3% confidence rating: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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