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```    1 ##DESCRIPTION
2 #  First Created: 3/2/02
4 #  Authors: Minock/Mueller
5 #  Math 162 Problems - Assignment 8 - Problem 12
6 #  Stewart 4th ed. Section 10.2
7 #  Find area of loop
8 ##ENDDESCRIPTION
9
10 ##KEYWORDS('parametric')
11 ## tcao tagged and PAID on 12-12-2003
12
13 ## DBsubject('Calculus')
14 ## DBchapter('Parametric Equations and Polar Coordinates')
15 ## DBsection('Tangents and Areas')
16 ## Date('6/3/2002')
17 ## Author('')
18 ## Institution('')
19 ## TitleText1('Calculus Early Transcendentals')
20 ## EditionText1('4')
21 ## AuthorText1('Stewart')
22 ## Section1('10.2')
23 ## Problem1('19')
24 DOCUMENT();
25
27 "PG.pl",
28 "PGbasicmacros.pl",
29 "PGchoicemacros.pl",
31 "PGauxiliaryFunctions.pl"
32 );
33
34 TEXT(&beginproblem);
36
37 \$a = random(1,5);
38 \$b = random(1,9);
39 \$c = random(1,9);
40 \$b2=\$b*\$b;
41 \$a2=\$a*\$a;
42
43 \$ans1 = \$b**2-(\$a**2)/3;
44 \$ans2 = 0;
45 \$ans3 = (8/15)*(\$a**5);
46 BEGIN_TEXT
47
48 Notice that the curve given by the parametric equations
49 \[ x=\$b2 - t^2      \]
50 \[    y= t^3 - \$a2 t     \]
51 is symetric about the x-axis.  (If t gives us the point (x,y), then -t will give
52 (x,-y) ).
53 \$BR
54 \$BR
55 At which x value is the tangent to this curve horizontal?
56 \$BR
57 At x =  \{ ans_rule(10) \}
58 END_TEXT
59 ANS(std_num_cmp(\$ans1));
60
61 BEGIN_TEXT
62 \$BR
63 At which t value is the tangent to this curve vertical?
64 \$BR
65 At t =  \{ ans_rule(10) \}
66 END_TEXT
67 ANS(std_num_cmp(\$ans2));
68
69 BEGIN_TEXT
70 \$BR The curve makes a loop which lies along the x-axis.  What is
71 the total area inside the loop?
72 \$BR
73 Area =  \{ ans_rule(40) \}
74 END_TEXT
75 ANS(std_num_cmp(\$ans3));
76
77 ENDDOCUMENT();
78
79
80
```