View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.2_Summing_an_Infinite_Series/10.2.48.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Limits and Derivatives')
    3 ## DBsection('Definition of the Derivative')
    4 ## KEYWORDS('calculus', 'derivatives', 'slope')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('10.2')
    9 ## Problem1('48')
   10 ## Author('Keith Thompson')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("Parser.pl");
   16 loadMacros("freemanMacros.pl");
   17 loadMacros("parserNumberWithUnits.pl");
   18 loadMacros("PGauxiliaryFunctions.pl");
   19 loadMacros("PGgraphmacros.pl");
   20 
   21 #$showPartialCorrectAnswers=1;
   22 $hin=random(8,15);
   23 $num=random(2,4,1);
   24 $m = random(1,2,1);
   25 $den=$m*$num+1;
   26 $den_num = $den-$num;
   27 $hin2=$hin*2;
   28 
   29 
   30 $ans1=$hin+$num*$hin*2/$den_num;
   31 Context()->texStrings;
   32 BEGIN_TEXT
   33 \{ beginproblem() \}
   34 \{ textbook_ref_exact("Rogawski ET 2e", "10.2","48") \}
   35 $PAR
   36 A ball dropped from a height of $hin feet begins to bounce. Each time it strikes  the ground, it returns to \(\frac{$num}{$den} \) of its previous height. What is the total distance traveled by the ball if it bounces infinitely many times?
   37 $PAR Total distance =  \{ans_rule()\}
   38 END_TEXT
   39 
   40 Context()->normalStrings;
   41 
   42 #ANS(Real($ans1)->cmp);
   43 
   44 ANS(NumberWithUnits($ans1,"ft")->cmp);
   45 Context()->texStrings;
   46 SOLUTION(EV3(<<'END_SOLUTION'));
   47 $PAR
   48 $SOL
   49 The ball initially drops $hin ft. Next the ball bounces back \(\frac{$num}{$den}\cdot $hin \) ft.  and then falls \(\frac{$num}{$den}\cdot $hin \) ft again. Repeating, the ball bounces back \(\frac{$num}{$den}\left(\frac{$num}{$den}\cdot $hin  \right)=\left(\frac{$num}{$den}\right)^2 $hin \) ft.  and then falls \(\left(\frac{$num}{$den}\right)^2 $hin \) ft again.
   50 Continuing this pattern, the total distance \(d\) traveled by the ball is given by the following infinite sum:
   51 \[d=$hin+2\cdot\frac{$num}{$den} $hin +2\cdot\left(\frac{$num}{$den}\right)^2 $hin+2\cdot\left(\frac{$num}{$den}\right)^3 $hin+\cdots\]
   52 \[ = $hin +2\cdot $hin\left(\frac{$num}{$den}\right)\left(1+\frac{$num}{$den}+\left(\frac{$num}{$den}\right)^2+\left(\frac{$num}{$den}\right)^3+\cdots\right)\]
   53 \[=$hin+$hin2 \left(\frac{$num}{$den}\right) \sum_{n=0}^\infty\left(\frac{$num}{$den}\right)^n.\]
   54 
   55 We use the formula for the sum of a geometric series to compute the sum of the resulting series:
   56 \[d=$hin + $hin2 \left(\frac{$num}{$den}\right)\left(\frac{1}{1-\frac{$num}{$den}}\right)=$hin+$hin2\left( \frac{$num}{$den_num}\right).\]
   57 Thus, the total distance the ball traveled is \($hin+$hin2\left( \frac{$num}{$den_num}\right)\) ft.
   58 
   59 END_SOLUTION
   60 ENDDOCUMENT();