document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1028733>Question 1028733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ball is thrown upward with an initial velocity of 50 feet per second. The height h (in feet) of the ball after t seconds is given by h=-16^2+50t. At the same time, a balloon is rising at a constant rate of 20 feet per second. Its height h in feet after seconds is given by h=20t. \r<BR>\n" );
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document.write( "a.) when do the ball and the balloon reach the same height?\r<BR>\n" );
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document.write( "b.) when does the ball reach its maximum height?\r<BR>\n" );
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document.write( "c.) when does the ball hit the ground?\r<BR>\n" );
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document.write( "Please help me, gladly appreciated <3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #644013 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=644013\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are to functions of time, and time is measured from the same moment taken at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > fro both functions. (That is not very clearly stated in the problem, but your teacher thinks it is understood to be that way).<BR>\n" );
document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B50t ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B50t\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a very realistic representation of what the height (in feet) of a ball would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds after it is kicked up (or shot up) from the ground. (A physics teacher would like that equation).<BR>\n" );
document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=20t ALIGN=MIDDLE  ALT=\"h%28t%29=20t\"   DISABLED_style= \"max-width:90%; height:auto;\" > must represent the height (in feet) of a balloon <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds after the ball is kicked/shot up. (I believe this is not very realistic for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E=0 ALIGN=MIDDLE  ALT=\"t%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and it would take some skill on the part of the balloon pilot to achieve that for some time).<BR>\n" );
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document.write( "a.) The ball and the balloon reach the same height when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20t=-16t%5E2%2B50t ALIGN=MIDDLE  ALT=\"20t=-16t%5E2%2B50t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t%5E2-50t%2B20t=0 ALIGN=MIDDLE  ALT=\"16t%5E2-50t%2B20t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16t%5E2-30t=0 ALIGN=MIDDLE  ALT=\"16t%5E2-30t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816t-30%29t=0 ALIGN=MIDDLE  ALT=\"%2816t-30%29t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Obviously, that would be true for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=0%2C%22or%22%2C16t=30%29 ALIGN=MIDDLE  ALT=\"system%28t=0%2C%22or%22%2C16t=30%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=0%2C%22or%22%2Ct=3%2F16=1.875%29 ALIGN=MIDDLE  ALT=\"system%28t=0%2C%22or%22%2Ct=3%2F16=1.875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the ball and the balloon reach the same height \"after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281.875seconds%29 ALIGN=MIDDLE  ALT=\"highlight%281.875seconds%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\".<BR>\n" );
document.write( "That means 1.875 seconds after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which should be the time the ball was kicked up (or shot up).\r<BR>\n" );
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document.write( "b.) The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B50t ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B50t\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic function that has a maximum for a certain value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that value represents when the ball reaches its maximum height.<BR>\n" );
document.write( "If you want to use a formula given in class, there is no need to think, you just apply the memorized formula.<BR>\n" );
document.write( "Just in case, the formula said the the maximum for a function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C0 ALIGN=MIDDLE  ALT=\"a%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > happens when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%222+a%22 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%222+a%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Of course, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B50t ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B50t\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but it is the same kind of unction, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-16%3C0 ALIGN=MIDDLE  ALT=\"a=-16%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=50 ALIGN=MIDDLE  ALT=\"b=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=0 ALIGN=MIDDLE  ALT=\"c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the maximum occurs when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28-50%29%2F%282%2A%28-16%29%29=50%2F32=1.5625 ALIGN=MIDDLE  ALT=\"t=%28-50%29%2F%282%2A%28-16%29%29=50%2F32=1.5625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The ball reaches maximum height after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281.5625seconds%29 ALIGN=MIDDLE  ALT=\"highlight%281.5625seconds%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Hey, the means that when the ball meets the balloon (at t=1.875 seconds), the ball is already on its way down.<BR>\n" );
document.write( "If you do not like to memorize stuff just because someone said so,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B50t ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B50t\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-h%2F16=t%5E2-%2850%2F16%29t ALIGN=MIDDLE  ALT=\"-h%2F16=t%5E2-%2850%2F16%29t\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-h%2F16=t%5E2-%2850%2F16%29t%2B%2850%2F32%29%5E2-%2850%2F32%29%5E2 ALIGN=MIDDLE  ALT=\"-h%2F16=t%5E2-%2850%2F16%29t%2B%2850%2F32%29%5E2-%2850%2F32%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-h%2F16=%28t-50%2F32%29%5E2-%2850%2F32%29%5E2 ALIGN=MIDDLE  ALT=\"-h%2F16=%28t-50%2F32%29%5E2-%2850%2F32%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16%28t-50%2F32%29%5E2%2B16%2850%2F32%29%5E2 ALIGN=MIDDLE  ALT=\"h=-16%28t-50%2F32%29%5E2%2B16%2850%2F32%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3C%3E20%2F32 ALIGN=MIDDLE  ALT=\"t%3C%3E20%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16%28t-50%2F32%29%5E2%3C0 ALIGN=MIDDLE  ALT=\"-16%28t-50%2F32%29%5E2%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16%28t-50%2F32%29%5E2%2B16%2850%2F32%29%5E2%3C16%2850%2F32%29%5E2 ALIGN=MIDDLE  ALT=\"h=-16%28t-50%2F32%29%5E2%2B16%2850%2F32%29%5E2%3C16%2850%2F32%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=20%2F32 ALIGN=MIDDLE  ALT=\"t=20%2F32\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16%28t-50%2F32%29%5E2=0 ALIGN=MIDDLE  ALT=\"-16%28t-50%2F32%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=16%2850%2F32%29%5E2=2500%2A16%2F32%5E2=2500%2F64=39.0625 ALIGN=MIDDLE  ALT=\"h=16%2850%2F32%29%5E2=2500%2A16%2F32%5E2=2500%2F64=39.0625\"   DISABLED_style= \"max-width:90%; height:auto;\" > has its maximum value.<BR>\n" );
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document.write( "c.) The ball is at groune level when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=0 ALIGN=MIDDLE  ALT=\"h=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16t%5E2%2B50t ALIGN=MIDDLE  ALT=\"h=-16t%5E2%2B50t\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%28-16t%2B50%29t ALIGN=MIDDLE  ALT=\"h=%28-16t%2B50%29t\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-16%28t-50%2F16%29t ALIGN=MIDDLE  ALT=\"h=-16%28t-50%2F16%29t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=0 ALIGN=MIDDLE  ALT=\"h=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=0%2C%22or%22%2Ct=16%2F50=3.125%29 ALIGN=MIDDLE  ALT=\"system%28t=0%2C%22or%22%2Ct=16%2F50=3.125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the ball leaves the ground at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and hits the ground after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283.125seconds%29 ALIGN=MIDDLE  ALT=\"highlight%283.125seconds%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "Here is a graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-0.5%2C3.5%2C-5%2C45%2Csqrt%2820x%2920x%2Fsqrt%2820x%29%2Csqrt%28-16x%5E2%2B50x%29%2A%28-16x%5E2%2B50x%29%2Fsqrt%28-16x%5E2%2B50x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "NOTE:<BR>\n" );
document.write( "Math is use every day (by a few people) to do the calculations they need to do in real life. Some people think they do not need math.<BR>\n" );
document.write( "Thirty-five years ago most people thought they did not need a microwave oven, because they did not know about them and they did not know how to use them. I spent a lot of money on one and enjoyed efficient cooking for the next 21 years. (Then it broke down and I got a new one). By then, most of those who said a microwave oven was not necessary had changed their mind. Would that ever happen with people who think you do not need math? </SPAN>\n" );
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