View of /trunk/NationalProblemLibrary/Michigan/Chap16Sec4/Q28.pg

    1 # DESCRIPTION
    2 # Problem from Calculus, multi-variable, Hughes-Hallett et al.,
    3 # originally from 5ed (with updates)
    4 # WeBWorK problem written by Gavin LaRose, <glarose@umich.edu>
    5 # ENDDESCRIPTION
    6 
    7 ## KEYWORDS('polar coordinates', 'integral', 'calculus')
    8 ## Tagged by glr 04/29/10
    9 
   10 
   11 ## DBsubject('Calculus')
   12 ## DBchapter('Multiple Integrals')
   13 ## DBsection('Double Integrals in Polar Coordinates')
   14 ## Date('')
   15 ## Author('Gavin LaRose')
   16 ## Institution('University of Michigan')
   17 ## TitleText1('Calculus')
   18 ## EditionText1('5')
   19 ## AuthorText1('Hughes-Hallett')
   20 ## Section1('16.4')
   21 ## Problem1('28')
   22 
   23 ## Textbook tags
   24 ## HHChapter1('Integrating Functions of Several Variables')
   25 ## HHSection1('Double Integrals in Polar Coordinates')
   26 
   27 DOCUMENT();
   28 
   29 loadMacros(
   30 "PGstandard.pl",
   31 "PGchoicemacros.pl",
   32 "MathObjects.pl",
   33 "parserNumberWithUnits.pl",
   34 # "parserFormulaWithUnits.pl",
   35 # "parserFormulaUpToConstant.pl",
   36 # "PGcourse.pl",
   37 );
   38 
   39 Context("Numeric");
   40 $showPartialCorrectAnswers = 1;
   41 
   42 $rad = random(2,7,1);
   43 $den = random(6,14,2);
   44 
   45 $mass = NumberWithUnits( "$den*pi*$rad^2/3", "g" );
   46 
   47 Context()->texStrings;
   48 TEXT(beginproblem());
   49 BEGIN_TEXT
   50 
   51 A disk of radius $rad cm has density $den g/cm\({}^2\)  at
   52 its center, density 0 at its edge, and its density is a linear
   53 function of the distance from the center. Find the mass of the disk.
   54 
   55 $PAR
   56 mass = \{ ans_rule(35) \}
   57 $BR
   58 ${BITALIC}(Include \{helpLink('units')\}.)$EITALIC
   59 
   60 END_TEXT
   61 Context()->normalStrings;
   62 
   63 ANS($mass->cmp() );
   64 
   65 ($dn,$dd) = reduce( $den,$rad );
   66 $dor = ( $dd==1 ) ? $dn : "\frac{$dn}{$dd}";
   67 ($rn,$rd) = reduce( $rad*$rad, 3 );
   68 $rsqo3 = ( $rd == 1 ) ? $rn : "\frac{$rn}{$rd}";
   69 ($an,$ad) = reduce( $den*$rad*$rad, 3 );
   70 $ans = ( $ad == 1 ) ? $an : "\frac{$an}{$ad}";
   71 
   72 Context()->texStrings;
   73 SOLUTION(EV3(<<'END_SOLUTION'));
   74 $PAR SOLUTION $PAR
   75 
   76 The density function is given by
   77 \[ \rho(r) = $dor($rad - r), \]
   78 where \(r\) is the distance from the center of the disk.
   79 So the mass of the disk in grams is
   80 \[
   81 \int_R \rho(r)\,dA = \int_0^{2\pi} \int_0^{$rad} $dor($rad - r)rdr\,d\theta
   82 = $dor \int_0^{2\pi} ($rado2\,r^2 - \frac13\,r^3)\bigg|_0^{$rad}\,d\theta
   83 \]
   84 \[
   85   = $dor \int_0^{2\pi} $rsqo3 \,d\theta
   86 = $ans \mbox{ g}.
   87 \]
   88 
   89 
   90 END_SOLUTION
   91 Context()->normalStrings;
   92 
   93 
   94 COMMENT('MathObject version');
   95 ENDDOCUMENT();