document.write( "Question 31653: Fireworks Problem...
\n" ); document.write( "t: elapsed time in seconds
\n" ); document.write( "v0: initial upward velocity in feet per second
\n" ); document.write( "h(t): altitude in feet\r
\n" ); document.write( "\n" ); document.write( "Let \"h%28t%29=+-16t%5E2%2Bv%5B0%5Dt\", where v0 is positive.
\n" ); document.write( "A. Use the disciminant to show that there are always two elapsed times at which altitude is 0.
\n" ); document.write( "B. Use your answer to part A or an analysis of quadratic functions to find the maximum altitude of the rocket.\r
\n" ); document.write( "\n" ); document.write( "Thank you! \n" ); document.write( "
Algebra.Com's Answer #18296 by mbarugel(146)\"\" \"About 
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\n" ); document.write( "The discriminant of a quadratic equation of the form \"A%2Ax%5E2+%2B+B%2Ax+%2B+C\" is\r
\n" ); document.write( "\n" ); document.write( "\"B%5E2+-4%2AA%2AC\"\r
\n" ); document.write( "\n" ); document.write( "In order for the equation to have two roots (ie, two values of x for which the result of the function is zero), this discriminant must be positive. In the quadratic equation you provide, we have:
\n" ); document.write( "A = -16
\n" ); document.write( "B = v0
\n" ); document.write( "C = 0\r
\n" ); document.write( "\n" ); document.write( "Therefore, the discriminant is:\r
\n" ); document.write( "\n" ); document.write( "\"%28v%5B0%5D%29%5E2+-4%2A%28-16%29%2A0+=+v%5B0%5D%5E2\"\r
\n" ); document.write( "\n" ); document.write( "Since v0 is greater than zero, then the disciminant is positive; so we know that the equation \"-16t%5E2+%2B+v%5B0%5Dt+=+0\" has two solutions. So we know that there are two values for t that make the altitude equal to 0.\r
\n" ); document.write( "\n" ); document.write( "When the quadratic coefficient in a quadratic equation is negative (in this case, it's -16), then the maximum of the equation can be found at its vertex, whose formula is \"-B%2F%282A%29\". So we know that at time \"t+=+-v%5B0%5D%2F%282%2A%28-16%29%29=v%5B0%5D%2F32\", the maximum altitude is attained.\r
\n" ); document.write( "\n" ); document.write( "In order to find the maximum altitude (what we've found is the TIME at which the maximum altitude is attained) we simply plug the t we found into the quadratic equation. So the maximum altitude will be:\r
\n" ); document.write( "\n" ); document.write( "\"-16%2A%28v%5B0%5D%2F32%29%5E2+%2B+%28v%5B0%5D%5E2%29%2F32\"
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