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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
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Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    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();
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();