## DBsubject('Calculus') ## DBchapter('Parametric Equations, Polar Coordinates, and Conic Sections') ## DBsection('Arc Length and Speed') ## KEYWORDS('calculus', 'parametric', 'polar', 'conic') ## TitleText1('Calculus: Early Transcendentals') ## EditionText1('2') ## AuthorText1('Rogawski') ## Section1('11.2') ## Problem1('18') ## Author('Christopher Sira') ## Institution('W.H.Freeman') DOCUMENT(); loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); loadMacros("PGchoicemacros.pl"); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); loadMacros("PGchoicemacros.pl"); $context = Context();$t = random(1, 15); $den = ($t**2 + 1)**2; $con =$t**2; $exp =$t * (4/$den + 9*$con)**.5; $ans = Formula("$exp"); Context()->texStrings; BEGIN_TEXT \{ beginproblem() \} \{ textbook_ref_exact("Rogawski ET 2e", "11.2","18") \} $PAR Determine the speed $$s(t)$$ of a particle with a given trajectory at a time $$t_0$$ (in units of meters and seconds). $c(t) = (\ln(t^2 + 1), t^3), \, t = t$$PAR \{ ans_rule() \} $PAR END_TEXT Context()->normalStrings; ANS($ans->cmp); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION')); $PAR$SOL We have $$x = \ln(t^2 + 1), \, y = t^3$$, so $$x' = \frac{2t}{t^2 + 1}$$ and $$y' = 3t^2$$. The speed of the particle at time t is thus $\frac{ds}{dt} = \sqrt{x'(t)^2 + y'(t)^2} = \sqrt{ \frac{4t^2}{ \left( t^2 + 1 \right) ^2 } + 9t^4 } = t \sqrt{\frac{4}{ \left( t^2 + 1 \right) ^2 } + 9t^2}$ The speed at time $$t = t$$ is $\frac{ds}{dt} \mid _{t = t} = t \sqrt{ \frac{4}{den} + 9 \cdot con } = ans$ END_SOLUTION ENDDOCUMENT();