# Problem10

Prep Main Page > Web Conference 2 > Sample Problems > Problem 10

This is Library/Rochester/setDerivatives10_5Optim/S04.07.Optimization.PTP09b.pg

```DOCUMENT();

#"PGbasicmacros.pl",
#"PGchoicemacros.pl",
#"PGauxiliaryFunctions.pl",
"PGstandard.pl",
"MathObjects.pl",
"parserNumberWithUnits.pl",
"parserFormulaWithUnits.pl",
);

TEXT(beginproblem());

Context("Numeric");

\$a = random(2,5,1);
\$b = random(6,12,1);
\$row = random(2,4,1);
\$walk = \$row + 1;

\$cp = (\$row*\$a)/((\$walk**2 - \$row**2)**(0.5));

\$function = FormulaWithUnits("(sqrt(\$a**2 + x**2))/\$row + (\$b - x)/\$walk","hr");
\$critical = NumberWithUnits("(\$row*\$a)/((\$walk^2 - \$row^2)^(0.5))","mi");
\$leasttravel = NumberWithUnits("((\$a^2 + \$cp^2)**(0.5))/\$row + (\$b - \$cp)/\$walk","hr");

Context()->texStrings;
BEGIN_TEXT
A small island is \$a miles from the nearest point P on the straight
shoreline of a large lake.  If a woman on the island can row a boat
\$row miles per hour and can walk \$walk miles per hour, where should
the boat be landed in order to arrive at a town \$b miles down the
shore from P in the least time?  Let \( x \) be the distance (in miles) between
point P and where the boat lands on the lakeshore.
\$BR
\$BR
(a) Enter a function \( T(x) \) that describes the total amount of
time the trip takes as a function of the distance \( x \).
\$BR
\( T(x) = \) \{ans_rule(30)\}
\$BR
\$BR
(b) What is the distance \( x = c \) that minimizes the travel time?
\$BR
\( c = \) \{ans_rule(25)\}
\$BR
\$BR
(c) What is the least travel time?
\$BR
The least travel time is \{ans_rule(25)\}
END_TEXT
Context()->normalStrings;
HINT(EV2(<<END_HINT));
When velocity is constant, time is distance divided by velocity.
END_HINT