View of /trunk/rochester_problib/setIntegrals23Work/ns6_5_12.pg

    1 #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.