## 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();