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| 1 : | sam | 2 | #DESCRIPTION |
| 2 : | # Integration | ||
| 3 : | # Application-based | ||
| 4 : | # Physics--Work. | ||
| 5 : | #ENDDESCRIPTION | ||
| 6 : | |||
| 7 : | #KEYWORDS('Integration', 'Physics', 'Applications') | ||
| 8 : | |||
| 9 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 10 : | |||
| 11 : | loadMacros( | ||
| 12 : | "PG.pl", | ||
| 13 : | "PGbasicmacros.pl", | ||
| 14 : | "PGchoicemacros.pl", | ||
| 15 : | "PGanswermacros.pl", | ||
| 16 : | "PGauxiliaryFunctions.pl" | ||
| 17 : | ); | ||
| 18 : | |||
| 19 : | TEXT(&beginproblem); | ||
| 20 : | $showPartialCorrectAnswers = 1; | ||
| 21 : | |||
| 22 : | #Here we ensure that the height of the pool is always greater than the depth of | ||
| 23 : | #the water | ||
| 24 : | $c1 = random(5,12,.5); | ||
| 25 : | $c2 = random(1,11,.5); | ||
| 26 : | @cs =($c1,$c2); | ||
| 27 : | @sortedcs = num_sort(@cs); | ||
| 28 : | $d = $sortedcs[0]; | ||
| 29 : | $h = $sortedcs[1]; | ||
| 30 : | $r = random(8,20,.5) ; | ||
| 31 : | $w = random(63,66,.1); | ||
| 32 : | $pi = 4*arctan(1); | ||
| 33 : | |||
| 34 : | BEGIN_TEXT | ||
| 35 : | You are visiting your friend Fabio's house. You find that, as a joke, he filled | ||
| 36 : | his swimming pool with Kool-Aid, which dissolved perfectly into the water. | ||
| 37 : | However, now that you want to swim, you must remove all of the Kool-Aid | ||
| 38 : | contaminated water. The swimming pool is round, with a $r foot radius. It is $h | ||
| 39 : | feet tall and has $d feet of water in it. $BR | ||
| 40 : | How much work is required to remove all of the water by pumping it over the | ||
| 41 : | side? | ||
| 42 : | Use the physical definition of work, and the fact that the weight of the | ||
| 43 : | Kool-Aid contaminated water is \( \sigma = $w lbs/ft^3 \) $BR | ||
| 44 : | \{ans_rule(45)\} | ||
| 45 : | END_TEXT | ||
| 46 : | |||
| 47 : | &HINT(EV3(<<'EOT')); | ||
| 48 : | $HINT $BR | ||
| 49 : | The formula for work is: $BR | ||
| 50 : | |||
| 51 : | \[\int_{a}^{b} Force * distance \] $BR | ||
| 52 : | |||
| 53 : | Where distance is the distance over which the force is exerted. | ||
| 54 : | EOT | ||
| 55 : | |||
| 56 : | &SOLUTION(EV3(<<'EOF')); | ||
| 57 : | $SOL $BR | ||
| 58 : | Consider a horizontal cross-section of the pool, with thickness \(dx\) if we | ||
| 59 : | consider the x-axis to be vertical, in the center of the pool. This is | ||
| 60 : | just a very short cylinder, so its volume is: $BR | ||
| 61 : | |||
| 62 : | \( dV = \pi r^2 dx \). $BR | ||
| 63 : | |||
| 64 : | We know r, the radius of the pool, is a constant, \(r= $r\). Now that we have | ||
| 65 : | the volume of an arbitrary cross-section of the water, we need to find the | ||
| 66 : | force which is exerted on the volume. That force is nothing more than the | ||
| 67 : | weight. The constant \( \sigma \) gives us weight-per-volume of the liquid. | ||
| 68 : | Therefore, by multipling the volume of the slice by \( \sigma \), we find: $BR | ||
| 69 : | |||
| 70 : | \( dF = \sigma \pi r^2 dx \) $BR | ||
| 71 : | |||
| 72 : | Since Work (W) is given by: $BR | ||
| 73 : | |||
| 74 : | \( W = Fx = F\int_{a}^{b}xdx \) $BR | ||
| 75 : | |||
| 76 : | in the case of a constant force F, all that remains is to find an expression | ||
| 77 : | for \( D\), the distance each slice of water is lifted. If we consider the top | ||
| 78 : | of the pool as x=0, $h-$d is the distance to the surface of the water, since the | ||
| 79 : | height of the pool is $h, and the depth is $d. So we have the distance | ||
| 80 : | x from x=$h-$d until x=$h. This results in the integral: $BR | ||
| 81 : | |||
| 82 : | \[ W = \int_{$h-$d}^{$h} \sigma \pi r^2 xdx \] $BR | ||
| 83 : | |||
| 84 : | which is simple to evaluate. | ||
| 85 : | EOF | ||
| 86 : | |||
| 87 : | $answer = (.5*$w*$pi*$r**2)*(2*$d*$h - $d**2) ; | ||
| 88 : | &ANS(std_num_cmp($answer)); | ||
| 89 : | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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