[rochester] / trunk / rochester_problib / setIntegrals23Work / ns6_5_12.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# Annotation of /trunk/rochester_problib/setIntegrals23Work/ns6_5_12.pg

 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.