jj2.pg

## DESCRIPTION
## Precalculus: Exponential and Logarithm Functions
## ENDDESCRIPTION

## Tagged by cmd6a 3/12/06

## DBsubject(Financial mathematics)
## DBchapter(Interest)
## DBsection(Compound interest)
## Institution(ASU)
## Level(4)

#
# First comes some stuff that appears at the beginning of every problem
#

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
  "PGstandard.pl",
  "PGchoicemacros.pl",
  "PGasu.pl",
  "PGgraphmacros.pl",
  "PGcourse.pl"
);

TEXT(beginproblem());

#
# Now we do the randomization of variables, and other computations
# as needed for this problem.  Sometimes we compute the answers here.
#

$a = random(3000,6000,1000);
$rt = random(6,9);
$b = random(0,1);
if($b) {
 ($tword, $t) = ("semiannually", 2);
} else {
  ($tword, $t) = ("quarterly", 4);
}

$targ = random(15, 40, 5);
$ans3 = ln($targ*1000/$a)/(ln(1+$rt/(100*$t))*$t);
$xd = 1.1*$ans3+10;
$xd = int($xd/10)+1;
$xd *= 10;
$xdl = -0.05*$ans3;
$ydl = -30*$targ;

$gr = init_graph($xdl, $ydl, $xd, $targ*1200, 'axes'=>[0,0]);
$gr2 = init_graph($xdl, $ydl, $xd, $targ*1200, 'axes'=>[0,0]);
$gr3 = init_graph($xdl, $ydl, $xd, $targ*1200, 'axes'=>[0,0]);
$gr4 = init_graph($xdl, $ydl, $xd, $targ*1200, 'axes'=>[0,0]);

$corrans = "$a*(1+$rt/(100*$t))^($t* x)";
$inc1 = "($a)*(1+$rt/(100*$t))^($t* x)-$a";
$inc2 = "$a*(1+$rt/(100*$t))^(-$t* x)";
$inc3 = "$a*(1+$rt/(100*$t))^($t* x*2)";

$f1 = FEQ("$corrans for x in <$xdl, $xd> using color:blue and weight:2");
($fig1) = plot_functions($gr, $f1);

($fig1) = plot_functions($gr2, 
FEQ("$inc1 for x in <$xdl, $xd> using color:blue and weight:2")
);

($fig1) = plot_functions($gr3, 
FEQ("$inc2 for x in <$xdl, $xd> using color:blue and weight:2")
);

($fig1) = plot_functions($gr4, 
FEQ("$inc3 for x in <$xdl, $xd> using color:blue and weight:2")
);
$lab = new Label(0, $a, "$a",'black','left');

$gr->lb($lab);
$gr2->lb($lab);
$gr3->lb($lab);
$gr4->lb($lab);
$mc = new_multiple_choice();
$mc->qa("Make a graph of \(A(t)\).  Which graph most closely resembles the graph of \(A(t)\)?".
"$BR $BBOLD Note:$EBOLD you can click on the graphs to get a larger view.$BR$BR".
"Each graph goes to $xd on the \(x\)-axis and ".($targ*1200)." on the \(y\)-axis.$BR$BR",
image(insertGraph($gr),
    alt=>"Graph of exponential growth starting at y = $a, curving upward with increasing steepness as x approaches $xd."
));

$mc->extra(image(insertGraph($gr2),
    alt=>"Exponential growth curve starting at y = $a, increasing and concave up as x increases to $xd."
),
image(insertGraph($gr3),
    alt=>"Graph of exponential growth function A(t) starting at initial value and increasing with upward concavity over time."
),
image(insertGraph($gr4),
    alt=>"Exponential growth curve starting at initial value, increasing and concave up, representing compound interest over time."
));

#
# Now the problem text itself, with ans_rule's to indicate where the
# answers go.  You can stop entering text, do more computations, and then
# start up again if you want.
#

BEGIN_TEXT
An amount of ${DOLLAR}$a is invested at an interest rate of $rt$PERCENT per
year, compounded $tword.
$BR$BR
Find the value \(A(t)\) of the investment after \(t\) years.
$BR$BR
\(A(t) = \) \{ans_rule(40)\} dollars
$BR$BR
$BBOLD Note:$EBOLD remember to express your function in the variable \(t\).
$BR$BR
\{ $mc->print_q() \}
\{ $mc->print_a() \}
$BR$BR
Use $BITALIC your$EITALIC graph of \(A(t)\) to determine when this investment
will amount to ${DOLLAR}$targ,000.  Give your answer in years, correct to the nearest
hundredth.  You will have to zoom on your graph to determine the answer.
$BR$BR
\{ans_rule(40)\} years
END_TEXT

#
# Tell WeBWork how to test if answers are right.  These should come in the
# same order as the answer blanks above.  You tell WeBWork both the type of
# "answer evaluator" to use, and the correct answer.
#

ANS(fun_cmp("$a*(1+$rt/(100*$t))^($t* t)", var=>['t']));
ANS(radio_cmp($mc->correct_ans));

ANS(num_cmp("ln($targ*1000/$a)/(ln(1+$rt/(100*$t))*$t)", tolType=>'absolute',
tol=>'0.0051'));

ENDDOCUMENT();        # This should be the last executable line in the problem.