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Tue Nov 8 15:17:41 2011 UTC (2 years, 5 months ago) by aubreyja
File size: 2059 byte(s)
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('Parametric Equations, Polar Coordinates, and Conic Sections')
3 ## DBsection('Arc Length and Speed')
4 ## KEYWORDS('calculus', 'parametric', 'polar', 'conic')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('11.2')
9 ## Problem1('16')
10 ## Author('Christopher Sira')
11 ## Institution('W.H.Freeman')
12
13 DOCUMENT();
19 $context = Context(); 20 21 22$a = random(2, 5);
23 $b = random(6, 9); 24$c = random(2, 9);
25
26 $ac =$a * $c; 27$bc = $b *$c;
28 $acsq =$ac**2;
29 $bcsq =$bc**2;
30 $csq =$c**2;
31 $q =$bcsq - $acsq; 32$acsqonc = $acsq /$csq;
33 $bcsqonc =$q / $csq; 34 35$exp = $c * ($acsqonc + $bcsqonc * (sin($c * (pi/4)))**2)**.5;
36
37 $ans = Formula("$exp");
38
39
40 Context()->texStrings;
41 BEGIN_TEXT
42 \{ beginproblem() \}
43 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","16") \}
44 $PAR 45 Determine the speed $$s(t)$$ of a particle with a given trajectory at a time $$t_0$$ (in units of meters and seconds). 46 $c(t) = (a \sin c t, b \cos c t), \, t = \frac{\pi}{4}$ 47$PAR
48 \{ ans_rule() \}
49 $PAR 50 END_TEXT 51 Context()->normalStrings; 52 53 ANS($ans->cmp);
54
55 Context()->texStrings;
56 SOLUTION(EV3(<<'END_SOLUTION'));
57 $PAR 58$SOL
59 We have $$x = a \sin c t, \, y = b \cos c t$$, hence $$x' = ac \cos c t, \, y' = - bc \sin c t$$.  Thus, the speed of the particle at time t is
60 $\frac{ds}{dt} = \sqrt{x'(t)^2 + y'(t)^2} = \sqrt{acsq \cos^2 c t + bcsq \sin^2 c t}$
61 $= \sqrt{acsq \left( \cos^2 c t + \sin^2 c t \right) + q \sin^2 c t} = c \sqrt{acsqonc + bcsqonc \sin^2 c t}$
62 Thus,
63 $\frac{ds}{dt} = c \sqrt{acsqonc + bcsqonc \sin^2 c t}$
64 The speed at time $$t = \frac{\pi}{4}$$ is thus
65 $\frac{ds}{dt} \mid _{t = \pi/4} = c \sqrt{acsqonc + bcsqonc \sin^2 \left( c \frac{\pi}{4} \right)} \cong ans$
66 END_SOLUTION
67
68 ENDDOCUMENT();