document.write( "Question 77426: For independence Day, the village of treetop is planning to launch fireworks from a barge on the lake. The rockets will achieve a height represented by the function H(t)=-16t^2+96t+4 where h(t) is measured in feet and t is time in seconds after launch.
\n" ); document.write( "A) At what time to the nearest second is the fireworks rocket at its highest point ?
\n" ); document.write( "B) if the rocket is designed to explode at its highest point, to the nearest foot, how high is the rocket when it explodes? \n" ); document.write( "
Algebra.Com's Answer #55534 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
a)
\n" ); document.write( "To find the time at the highest point, lets complete the square for \"H%28t%29=-16t%5E2%2B96t%2B4+\" so we can convert it into vertex form:\r
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Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

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\n" ); document.write( " \"y=-16+x%5E2%2B96+x%2B4\" Start with the given equation
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\n" ); document.write( " \"y-4=-16+x%5E2%2B96+x\" Subtract \"4\" from both sides
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\n" ); document.write( " \"y-4=-16%28x%5E2-6x%29\" Factor out the leading coefficient \"-16\"
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\n" ); document.write( " Take half of the x coefficient \"-6\" to get \"-3\" (ie \"%281%2F2%29%28-6%29=-3\").
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\n" ); document.write( " Now square \"-3\" to get \"9\" (ie \"%28-3%29%5E2=%28-3%29%28-3%29=9\")
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\n" ); document.write( " \"y-4=-16%28x%5E2-6x%2B9-9%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"9\" does not change the equation
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\n" ); document.write( " \"y-4=-16%28%28x-3%29%5E2-9%29\" Now factor \"x%5E2-6x%2B9\" to get \"%28x-3%29%5E2\"
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\n" ); document.write( " \"y-4=-16%28x-3%29%5E2%2B16%289%29\" Distribute
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\n" ); document.write( " \"y-4=-16%28x-3%29%5E2%2B144\" Multiply
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\n" ); document.write( " \"y=-16%28x-3%29%5E2%2B144%2B4\" Now add \"4\" to both sides to isolate y
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\n" ); document.write( " \"y=-16%28x-3%29%5E2%2B148\" Combine like terms
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\n" ); document.write( " Now the quadratic is in vertex form \"y=a%28x-h%29%5E2%2Bk\" where \"a=-16\", \"h=3\", and \"k=148\". Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.
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\n" ); document.write( " Check:
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\n" ); document.write( " Notice if we graph the original equation \"y=-16x%5E2%2B96x%2B4\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-16x%5E2%2B96x%2B4%29\" Graph of \"y=-16x%5E2%2B96x%2B4\". Notice how the vertex is (\"3\",\"148\").
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\n" ); document.write( " Notice if we graph the final equation \"y=-16%28x-3%29%5E2%2B148\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-16%28x-3%29%5E2%2B148%29\" Graph of \"y=-16%28x-3%29%5E2%2B148\". Notice how the vertex is also (\"3\",\"148\").
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\n" ); document.write( " So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.
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\n" ); document.write( "\n" ); document.write( "So the vertex is (3,148) where the first coordinate is t, so t=3. \r
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\n" ); document.write( "\n" ); document.write( "b)
\n" ); document.write( "Since the rocket explodes at the vertex (the highest point), the height of the explosion is 148 ft (ie it's the y coordinate of the vertex). \n" ); document.write( "
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