document.write( "Question 998202: Anne is standing on a straight road and wants to reach her helicopter which is located 2 miles down the road for her and a mile off the road in a field. she can run 5 miles an hour on the road and 3 miles per hour in the field. she plans to run down the road, then cut diagonally across the field to reach the helicopter.
\n" ); document.write( "where should she leave the road in order to reach the helicopter in exactly 42 minutes (0.7 hour)
\n" ); document.write( "where should she leave the road in order to reach the helicopter as soon as possible.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #616046 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let's say Anne runs at 5 mph down the road,
\n" ); document.write( "until she is $\"x\"$ miles before the point directly in front of the helicopter.
\n" ); document.write( "Then, she runs across the field in a straight line towards the helicopter.
\n" ); document.write( "The road, field, helicopter, and Anne's path look like this:

\n" ); document.write( "So Anne runs on the road for $\"2-x\"$ miles at $\"5\"$ mph in $\"%282-x%29%2F5\"$ hours.
\n" ); document.write( "Then, she runs across the field for $\"sqrt%28x%5E2%2B1%29\"$ miles at $\"3\"$ mph in $\"sqrt%28x%5E2%2B1%29%2F3\"$ hours.
\n" ); document.write( "Anne's total time as a function of $\"x\"$ is
\n" ); document.write( " $\"t%28x%29=%282-x%29%2F5%2Bsqrt%28x%5E2%2B1%29%2F3=%286-3x%2B5sqrt%28x%5E2%2B1%29%29%2F15\"$ hours.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If Anne wants to reach the helicopter in exactly 42 minutes (0.7 hour), $\"x\"$ must be such that
\n" ); document.write( " $\"%286-3x%2B5sqrt%28x%5E2%2B1%29%29%2F15=0.7\"$
\n" ); document.write( " $\"6-3x%2B5sqrt%28x%5E2%2B1%29=15%2A0.7\"$
\n" ); document.write( " $\"6-3x%2B5sqrt%28x%5E2%2B1%29=10.5\"$
\n" ); document.write( " $\"5sqrt%28x%5E2%2B1%29=10.5-6%2B3x\"$
\n" ); document.write( " $\"5sqrt%28x%5E2%2B1%29=4.5%2B3x\"$
\n" ); document.write( " $\"25%28x%5E2%2B1%29=%284.5%2B3x%29%5E2\"$
\n" ); document.write( " $\"25x%5E2%2B25=20.25%2B27x%2B9x%5E2\"$
\n" ); document.write( " $\"25x%5E2-9x%5E2-27x%2B25-20.25=0\"$
\n" ); document.write( " $\"16x%5E2-27x%2B4.75=0\"$
\n" ); document.write( " $\"x+=+%28-%28-27%29+%2B-+sqrt%28%28-27%29%5E2-4%2A16%2A4.75+%29%29%2F%282%2A16%29+\"$
\n" ); document.write( " $\"x+=+%2827+%2B-+sqrt%28729-304%29%29%2F32\"$
\n" ); document.write( " $\"x+=+%2827+%2B-+sqrt%28425%29%29%2F32\"$
\n" ); document.write( "The two solutions are:
\n" ); document.write( " $\"x%5B1%5D=about0.20\"$ (rounded), meaning Anne runs
\n" ); document.write( " $\"2-0.2=highlight%281.8%29\"$ miles down the road, and then leaves the road, and
\n" ); document.write( " $\"x%5B2%5D=about1.49\"$ (rounded), meaning Anne runs
\n" ); document.write( " $\"2-1.49=highlight%280.51%29\"$ miles down the road, and then leaves the road.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If Anne wants to reach the helicopter as soon as possible, we must chose $\"x\"$ to make $\"t%28x%29\"$ minimum.
\n" ); document.write( "I cannot think of another way other than resorting to calculus,
\n" ); document.write( "unless you are expected to use a graphing calculator (or a spreadsheet) to get an approximate answer.
\n" ); document.write( "The derivative of

is
\n" ); document.write( "

\n" ); document.write( "

\n" ); document.write( "When $\"t%28x%29\"$ is minimum, $\"dt%2Fdx%29=0\"$ , meaning that
\n" ); document.write( " $\"-3sqrt%28x%5E2%2B1%29%2B5x=0\"$ --> $\"5x=3sqrt%28x%5E2%2B1%29\"$ --> $\"25x%5E2=9%28x%5E2%2B1%29\"$ --> $\"25x%5E2=9x%5E2%2B9\"$ --> $\"25x%5E2-9x%5E2=9\"$ --> $\"16x%5E2=9\"$ --> $\"x%5E2=9%2F16\"$ --> $\"x=3%2F4=0.75\"$
\n" ); document.write( "So, if Anne wants to reach the helicopter as soon as possible,
\n" ); document.write( "she should leave the road after running (((2-0.75=highlight(1.25)}}} miles. \n" ); document.write( "

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