document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198646>Question 1198646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4.A model rocket is shot straight up from the roof of a school. The height, h, in meters, after t seconds can be approximated by h(t)=-5t2+22t+15. Answer the following questions algebraically, showing all of your work. \r<BR>\n" );
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document.write( "What is the height of the school? <BR>\n" );
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document.write( "When does the rocket hit the ground? \r<BR>\n" );
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document.write( "What is the maximum height of the rocket?\r<BR>\n" );
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document.write( "How long does it take for the rocket to pass a window that is 10m above the ground? (hint: make h=10, group like terms, then solve the quadratic equation)]\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832268 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=832268\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll break the four questions into part (a) through part (d).\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "The rocket starts on the roof at time t = 0.<BR>\n" );
document.write( "Plug this value into the equation to get<BR>\n" );
document.write( "h(t) = -5t^2+22t+15<BR>\n" );
document.write( "h(0) = -5(0)^2+22(0)+15<BR>\n" );
document.write( "h(0) = 15\r<BR>\n" );
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document.write( "The school's height is 15 meters.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "The rocket hits the ground at a height of h = 0.<BR>\n" );
document.write( "We'll replace h(t) with 0 and apply the quadratic formula to solve.\r<BR>\n" );
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document.write( "h(t) = -5t^2+22t+15<BR>\n" );
document.write( "0 = -5t^2+22t+15<BR>\n" );
document.write( "-5t^2+22t+15 = 0\r<BR>\n" );
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document.write( "Plug in a = -5, b = 22, c = 15<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"t+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22%2B-sqrt%28%2822%29%5E2-4%28-5%29%2815%29%29%29%2F%282%28-5%29%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22%2B-sqrt%28%2822%29%5E2-4%28-5%29%2815%29%29%29%2F%282%28-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22%2B-sqrt%28784%29%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22%2B-sqrt%28784%29%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22%2B-++28%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22%2B-++28%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22%2B28%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22%2B28%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22-28%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22-28%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%286%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%286%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-50%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-50%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-0.6 ALIGN=MIDDLE  ALT=\"t+=+-0.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+5 ALIGN=MIDDLE  ALT=\"t+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The negative time value doesn't make much sense, so we ignore it.<BR>\n" );
document.write( "t = 5 is the only practical solution here. \r<BR>\n" );
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document.write( "At the time stamp of 5 seconds is when the rocket hits the ground.\r<BR>\n" );
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document.write( "Note: The rocket does not fall straight down or else it would hit the roof of the school, rather than the ground. <BR>\n" );
document.write( "In other words, the rocket must drift to one side as it falls.\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "I'll replace t with x, and replace h with y.\r<BR>\n" );
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document.write( "The equation we have is y = -5x^2+22x+15<BR>\n" );
document.write( "Which when compared to y = ax^2+bx+c we have<BR>\n" );
document.write( "a = -5<BR>\n" );
document.write( "b = 22<BR>\n" );
document.write( "c = 15\r<BR>\n" );
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document.write( "Plug the 'a' and 'b' values into this formula<BR>\n" );
document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -22/(2(-5))<BR>\n" );
document.write( "h = 2.2<BR>\n" );
document.write( "This is the x coordinate of the vertex (h,k)\r<BR>\n" );
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document.write( "Then use that to find the value of the y coordinate of the vertex.<BR>\n" );
document.write( "y = -5x^2+22x+15<BR>\n" );
document.write( "y = -5(2.2)^2+22(2.2)+15<BR>\n" );
document.write( "y = 39.2\r<BR>\n" );
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document.write( "The vertex is located at (2.2, 39.2) which is the highest point of the parabola.<BR>\n" );
document.write( "The rocket reaches its max height of 39.2 meters at exactly the timestamp of 2.2 seconds.\r<BR>\n" );
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document.write( "Part (d)\r<BR>\n" );
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document.write( "This is similar to the initial approach taken in part (b). <BR>\n" );
document.write( "This time we replace h with 10 instead of 0. The quadratic formula will show up again.\r<BR>\n" );
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document.write( "h(t) = -5t^2+22t+15<BR>\n" );
document.write( "10 = -5t^2+22t+15<BR>\n" );
document.write( "0 = -5t^2+22t+15-10<BR>\n" );
document.write( "0 = -5t^2+22t+5\r<BR>\n" );
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document.write( "Now turn to the quadratic formula.<BR>\n" );
document.write( "This time we plug in: <BR>\n" );
document.write( "a = -5<BR>\n" );
document.write( "b = 22<BR>\n" );
document.write( "c = 5<BR>\n" );
document.write( "to get the following<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22%2B-++24.166092%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22%2B-++24.166092%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22%2B24.166092%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22%2B24.166092%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-22-24.166092%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-22-24.166092%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%282.166092%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%282.166092%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-46.16609%29%2F%28-10%29 ALIGN=MIDDLE  ALT=\"t+=+%28-46.16609%29%2F%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-0.216609 ALIGN=MIDDLE  ALT=\"t+=+-0.216609\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+4.616609 ALIGN=MIDDLE  ALT=\"t+=+4.616609\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The decimal values are approximate.<BR>\n" );
document.write( "Like earlier we ignore the negative result.\r<BR>\n" );
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document.write( "The rocket reaches a height of exactly 10 meters at the timestamp of roughly 4.616609 seconds.\r<BR>\n" );
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document.write( "Side note: All of this ignores air resistance which greatly complicates the problem. <BR>\n" );
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