document.write( "Question 861872: Solve using derivatives - A dart is shot straight up from 1.5 metres above ground level. The distance (d), in metres, the dart is above the ground at time (t), in seconds is: \r
\n" ); document.write( "\n" ); document.write( " \"d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5\"\r
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\n" ); document.write( "\n" ); document.write( "1) What is the maximum height the dart reaches? I think this is 21.92 m
\n" ); document.write( "2) for how many seconds is the dart in the air?
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Algebra.Com's Answer #519376 by Alan3354(69443)\"\" \"About 
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Solve using derivatives - A dart is shot straight up from 1.5 metres above ground level. The distance (d), in metres, the dart is above the ground at time (t), in seconds is: \r
\n" ); document.write( "\n" ); document.write( " \"d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5\"\r
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\n" ); document.write( "\n" ); document.write( "1) What is the maximum height the dart reaches? I think this is 21.92 m
\n" ); document.write( "Already answered.
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\n" ); document.write( "2) for how many seconds is the dart in the air?
\n" ); document.write( "From t = 0 until d(t) = 0 at impact.
\n" ); document.write( " \"d%28t%29=+-4.9t%5E2+%2B+20t+%2B+1.5+=+0\"
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Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation \"ax%5E2%2Bbx%2Bc=0\" (in our case \"-4.9x%5E2%2B20x%2B1.5+=+0\") has the following solutons:
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\n" ); document.write( " \"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"
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\n" ); document.write( " For these solutions to exist, the discriminant \"b%5E2-4ac\" should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant \"b%5E2-4ac\": \"b%5E2-4ac=%2820%29%5E2-4%2A-4.9%2A1.5=429.4\".
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\n" ); document.write( " Discriminant d=429.4 is greater than zero. That means that there are two solutions: \"+x%5B12%5D+=+%28-20%2B-sqrt%28+429.4+%29%29%2F2%5Ca\".
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\n" ); document.write( " \"x%5B1%5D+=+%28-%2820%29%2Bsqrt%28+429.4+%29%29%2F2%5C-4.9+=+-0.0736703080011585\"
\n" ); document.write( " \"x%5B2%5D+=+%28-%2820%29-sqrt%28+429.4+%29%29%2F2%5C-4.9+=+4.15530296106238\"
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\n" ); document.write( " Quadratic expression \"-4.9x%5E2%2B20x%2B1.5\" can be factored:
\n" ); document.write( " \"-4.9x%5E2%2B20x%2B1.5+=+%28x--0.0736703080011585%29%2A%28x-4.15530296106238%29\"
\n" ); document.write( " Again, the answer is: -0.0736703080011585, 4.15530296106238.\n" ); document.write( "Here's your graph:
\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-4.9%2Ax%5E2%2B20%2Ax%2B1.5+%29\"

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\n" ); document.write( "Ignore the negative solution.
\n" ); document.write( "t =~ 4.155 seconds
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