document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=115783>Question 115783</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0072934824/algebrahomeworkh>Algebra For college students</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> A projectile is thrown upward so that the its distance above the ground after t seconds is h(t)=-16t^2+308t<BR>\n" );
document.write( "After how many seconds does it reach its maximum height?\r<BR>\n" );
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document.write( "What is that maximum height?\r<BR>\n" );
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document.write( "The area of a square is numerically 12 more than the perimeter. Find the length of the side?\r<BR>\n" );
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document.write( "These two problems are really hard can anyone help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84253 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=84253\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First problem:<BR>\n" );
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document.write( "You are given the equation of the height of an object that is thrown upward as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=-16t%5E2%2B308t ALIGN=MIDDLE  ALT=\"h%28t%29=-16t%5E2%2B308t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the first question you are asked is to find the time that the object reaches its maximum<BR>\n" );
document.write( "height. The object starts out at ground level, rises to a peak, and then falls back to <BR>\n" );
document.write( "ground level. Neglecting air resistance and other minor considerations, it spends half its<BR>\n" );
document.write( "time rising and half its time falling back to ground level. So one way you can find the<BR>\n" );
document.write( "time it takes to reach its peak height is to find the time of launch and the total time that<BR>\n" );
document.write( "goes by until it hits the ground ... then divide that time by 2. <BR>\n" );
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document.write( "Notice that at ground level the height given by h(t) is zero. So let's substitute zero<BR>\n" );
document.write( "into the equation for h(t) and get (after reversing the sides of the equation to get it <BR>\n" );
document.write( "into a little more familiar form):<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t%5E2+%2B+308t+=+0 ALIGN=MIDDLE  ALT=\"-16t%5E2+%2B+308t+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that t is a common factor of both the terms on the left side, so it can be factored to<BR>\n" );
document.write( "make the left side become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2A%28-16t%2B308%29+=+0 ALIGN=MIDDLE  ALT=\"t%2A%28-16t%2B308%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that this equation will be true if either of the factors is equal to zero because a <BR>\n" );
document.write( "multiplication by zero on the left side will make the left side equal the zero on the right<BR>\n" );
document.write( "side.<BR>\n" );
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document.write( "Setting the first factor [which is t] equal to zero results in:<BR>\n" );
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document.write( "<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;\" ><BR>\n" );
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document.write( "This means that at t = 0 seconds the object is at ground launch. No surprise here.<BR>\n" );
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document.write( "Next setting the second factor [which is -16t + 308] equal to zero gives:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t+%2B+308+=+0 ALIGN=MIDDLE  ALT=\"-16t+%2B+308+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve this by first subtracting 308 from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16t+=+-308 ALIGN=MIDDLE  ALT=\"-16t+=+-308\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and then dividing both sides by -16 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-308%2F-16+=+%2B19.25 ALIGN=MIDDLE  ALT=\"t+=+-308%2F-16+=+%2B19.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells us that from time of launch at t = 0, 19.25 seconds later the object hits the ground.<BR>\n" );
document.write( "Since half of that time was spent rising and half of that time was spent falling back down, <BR>\n" );
document.write( "the time at which the object reaches its peak is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.25%2F2+=+9.625 ALIGN=MIDDLE  ALT=\"19.25%2F2+=+9.625\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So at 9.625 seconds <BR>\n" );
document.write( "after launch the object is at its maximum height. <BR>\n" );
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document.write( "Another way you can find this time is to apply part of the quadratic formula. Recall that <BR>\n" );
document.write( "the quadratic formula applies to quadratic equations of the form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=at%5E2+%2B+bt+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"at%5E2+%2B+bt+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you compare this with your height equation you will see that a = -16, b = 308, and c = 0.<BR>\n" );
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document.write( "Then just use that portion of the quadratic formula that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-b%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"t+=+-b%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the <BR>\n" );
document.write( "time at the peak. Substituting 308 for b and -16 for a results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-308%29%2F%282%2A%28-16%29%29+=+-308%2F-32+=+9.625 ALIGN=MIDDLE  ALT=\"t+=+%28-308%29%2F%282%2A%28-16%29%29+=+-308%2F-32+=+9.625\"   DISABLED_style= \"max-width:90%; height:auto;\" > seconds.<BR>\n" );
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document.write( "This is the same answer, just a little different way of getting it. <BR>\n" );
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document.write( "Then to find the maximum height, just substitute 9.625 seconds for t in the height equation<BR>\n" );
document.write( "to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+-16%2A%289.625%29%5E2+%2B+308%2A%289.625%29+=+-16%2A%2892.640625%29+%2B+2964.5+=+-1482.25+%2B+2964.5+=+1482.25 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the object rises to a height of 1482.25 feet. That's quite a throw!!!! Check your problem<BR>\n" );
document.write( "to see if the 308 is the correct multiplier of the t term. If it is, that's really the<BR>\n" );
document.write( "answer .... 1482.25 feet up and 1482.25 feet back down again.<BR>\n" );
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document.write( "Next problem.<BR>\n" );
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document.write( "The Area of a square that has S as the length of one side is given by the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+S%5E2 ALIGN=MIDDLE  ALT=\"A+=+S%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the Perimeter of the square is the sum of the lengths of all its sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+S+%2B+S+%2B+S+%2B+S+=+4%2AS ALIGN=MIDDLE  ALT=\"P+=+S+%2B+S+%2B+S+%2B+S+=+4%2AS\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since the Area is 12 more than the Perimeter, if you take 12 away from the Area, the result<BR>\n" );
document.write( "will equal the Perimeter. In equation form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5E2+-+12+=+4S ALIGN=MIDDLE  ALT=\"S%5E2+-+12+=+4S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get this into the standard quadratic form by subtracting 4S from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5E2+-+4S+-+12+=+0 ALIGN=MIDDLE  ALT=\"S%5E2+-+4S+-+12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation factors to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28S+%2B2%29%28S+-+6%29+=+0 ALIGN=MIDDLE  ALT=\"%28S+%2B2%29%28S+-+6%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "As in the previous problem, this equation will be true if either factor equals zero. <BR>\n" );
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document.write( "Setting the first factor equal to zero results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"S+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 2 from both sides and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+-2 ALIGN=MIDDLE  ALT=\"S+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This answer doesn't make sense ... a side of minus 2 length??? Ignore it.<BR>\n" );
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document.write( "Setting the second factor equal to zero gives:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+-+6+=+0 ALIGN=MIDDLE  ALT=\"S+-+6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Add 6 to both sides and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+6 ALIGN=MIDDLE  ALT=\"S+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This looks better. A square with a side of 6 has an area of 6^2 = 36 and a perimeter of<BR>\n" );
document.write( "6*4 = 24. The area is 12 more than the perimeter, just as the problem specified. So the <BR>\n" );
document.write( "side length you were to find is S = 6.<BR>\n" );
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document.write( "Hope this helps with your understanding of these two problems. It's pretty late so you had<BR>\n" );
document.write( "better use your calculator to check my math. The process is correct, but I may have let<BR>\n" );
document.write( "a calculation error slip in ... I don't think so but better safe than sorry ...<BR>\n" );
document.write( ".\r<BR>\n" );
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document.write( " </SPAN>\n" );
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