View of /trunk/NationalProblemLibrary/Rochester/setDerivatives8RelatedRates/s2_8_21_mo.pg

    1 #DESCRIPTION
    2 #KEYWORDS('derivatives', 'related rates')
    3 #TYPE('word problem')
    4 # Related rates -- gravel dumped in a pile in a shape of a right circular
    5 #  cone with diameter = height.  Find the rate at which the height is growing
    6 #  given the volume, height and rate of change of the volume
    7 #ENDDESCRIPTION
    8 
    9 ##KEYWORDS('Derivatives')
   10 ##Tagged by ynw2d
   11 
   12 ## Modified ('06/20/2008')
   13 ## ModifiedBy('nbennett')
   14 
   15 DOCUMENT();        # This should be the first executable line in the problem.
   16 
   17 loadMacros(
   18 "PG.pl",
   19 "PGbasicmacros.pl",
   20 "PGanswermacros.pl",
   21 "PGauxiliaryFunctions.pl",
   22 "MathObjects.pl",
   23 "PGcourse.pl"         # Customization file for the course
   24 );
   25 
   26 TEXT(beginproblem());
   27 ########################################
   28 # Setup
   29 
   30 Context("Numeric");
   31 $showPartialCorrectAnswers = 1;
   32 
   33 Context()->variables->add(y=>'Real');
   34 
   35 Context()->flags->set(reduceConstants=>0);
   36 Context()->flags->set(reduceConstantFunctions=>0);
   37 
   38 
   39 $a = random(10,50,10);
   40 $h = random(10,25,1);
   41 
   42 $rate = Compute("$a*12/(3*($h^2))");
   43 
   44 ########################################
   45 # Main Text
   46 
   47 Context()->texStrings;
   48 BEGIN_TEXT
   49 Gravel is being dumped from a conveyor belt at a rate of \($a\) cubic feet per minute.
   50 It forms a pile in the shape of a right circular cone whose base diameter and height
   51 are always the same.  How fast is the height of the pile increasing when the pile is
   52 \($h\) feet high? Recall that the  volume of a right circular cone with height \(h\)
   53 and radius of the base \(r\) is given by \(V= \frac{1}{3}\pi r^2h \).
   54 $BR
   55 \{ans_rule(20) \}
   56 END_TEXT
   57 Context()->normalStrings;
   58 
   59 ########################################
   60 # Answers
   61 
   62 ANS($rate->cmp);
   63 
   64 ########################################
   65 # Solution
   66 
   67 Context()->texStrings;
   68 SOLUTION(EV3(<<'END_SOLUTION'));
   69 We are given that the diameter and height are always the same, so \(2r=h\) or
   70 \(r=\frac{h}{2}\).
   71 $BR $BR The equation \( V= \frac{1}{3}\pi r^2h \) can be rewritten as
   72 \( V= \frac{1}{3}\pi \left( \frac{h}{2} \right)^2h \) or \( V= \frac{1}{12}\pi h^3\).
   73 $BR $BR Differentiate both sides of the last equation with respect to time \((t)\):
   74 $BR $BR
   75 \(V' = \frac{1}{12}\pi 3 h^2 h'\)
   76 $BR $BR
   77 \(V' = \frac{1}{12} \pi 3 h^2 h'\)
   78 $BR $BR
   79 If \(V'=$a\) and \(h=$h\), then
   80 $BR $BR
   81 \($a = \frac{1}{12} \pi (3) $h^2 h'\)
   82 $BR $BR
   83 \(\displaystyle h' = \frac{$a(12)}{(3)$h^2\pi}\)
   84 END_SOLUTION
   85 Context()->normalStrings;
   86 
   87 ########################################
   88 
   89 ENDDOCUMENT();        # This should be the last executable line in the problem.