document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=470978>Question 470978</A>: <I><SPAN STYLE=\"word-break: break-all;\"> h(t)= c-(d-4t)^2; At time t=0, an initial height of 6 feet.  Until the ball hit the ground, its height in feet after t seconds was given by the function h (at the beginning of problem), in which c and d are positive constants.  If the ball reached its maximum height of 106 feet at time t=2.5, what was the height in feet of the ball at t=1 ?\r<BR>\n" );
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document.write( "The answer is 70, but I do not understand how they came up with this answer....please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #323064 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=323064\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is an exercise in finding the constants based on several pieces of knowledge that is given in the problem.  The use of c and d as constants without any explanation of what they actually are is an uncommon practice. I understand the math, but don't understand why the originator of the problem made it difficult by using an uncommon exercise. No wonder nobody has attempted this.  My complaint is over. On to the problem.<BR>\n" );
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document.write( "The equation you were given is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=+c-%28d-4t%29%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29=+c-%28d-4t%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's start by expanding the right side. Do that by squaring the term in the parenthesis 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=+c-%28d%5E2+-+8dt+%2B+16t%5E2%29 ALIGN=MIDDLE  ALT=\"h%28t%29=+c-%28d%5E2+-+8dt+%2B+16t%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Remove the parenthesis by changing the signs of all the terms within 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+=+c-d%5E2+%2B+8dt+-+16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29+=+c-d%5E2+%2B+8dt+-+16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that the first two terms on the right side of the equal sign are constants that do not depend on time.<BR>\n" );
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document.write( "The problem says that at t = 0, the height of the ball is 6 feet.  Therefore, if we set the right side equal to 6 and t = 0 we can learn something. Let's do that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c-d%5E2+%2B+8dt+-+16t%5E2+=+6 ALIGN=MIDDLE  ALT=\"c-d%5E2+%2B+8dt+-+16t%5E2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute zero as the value of t and the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c-d%5E2+%2B+0+%2B+0+=+6 ALIGN=MIDDLE  ALT=\"c-d%5E2+%2B+0+%2B+0+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we know that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+-+d%5E2+=+6 ALIGN=MIDDLE  ALT=\"c+-+d%5E2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Remember this. You are also told that when t = 2.5 seconds the height of the ball is 106 feet.  Let's now go back to the equation for h(t) and let h(2.5) = 106 and t = 2.5.  The equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+c-d%5E2+%2B+8dt+-+16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29+=+c-d%5E2+%2B+8dt+-+16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=106+=+c+-+d%5E2+%2B+8d%282.5%29+-+16%282.5%29%5E2 ALIGN=MIDDLE  ALT=\"106+=+c+-+d%5E2+%2B+8d%282.5%29+-+16%282.5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Remember now that we found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c-d%5E2%7D=+6 ALIGN=MIDDLE  ALT=\"c-d%5E2%7D=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So substitute 6 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c-d%5E2 ALIGN=MIDDLE  ALT=\"c-d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to make the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=106+=+6+%2B+8d%282.5%29+-+16%282.5%29%5E2 ALIGN=MIDDLE  ALT=\"106+=+6+%2B+8d%282.5%29+-+16%282.5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the last two terms on the right side:  <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%282.5%29=+20d ALIGN=MIDDLE  ALT=\"8d%282.5%29=+20d\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%282.5%29%5E2+=+100 ALIGN=MIDDLE  ALT=\"16%282.5%29%5E2+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute these values into the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=106+=+6+%2B+20d+-+100 ALIGN=MIDDLE  ALT=\"106+=+6+%2B+20d+-+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for d by first adding 100 to both sides of the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=206+=+6+%2B+20d ALIGN=MIDDLE  ALT=\"206+=+6+%2B+20d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and then subtracting 6 from both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200+=+20d ALIGN=MIDDLE  ALT=\"200+=+20d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally, divide both sides by 20 and you have d = 10.<BR>\n" );
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document.write( "Let's not lose sight of solving the problem for the height when t = 1.  Let's return to the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+%28c-d%5E2%29+%2B+8dt+-+16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29+=+%28c-d%5E2%29+%2B+8dt+-+16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We now know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+-+d%5E2+=+6 ALIGN=MIDDLE  ALT=\"c+-+d%5E2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+10 ALIGN=MIDDLE  ALT=\"d+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute these two values into the equation and it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+6+%2B+8%2810%29t+-+16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29+=+6+%2B+8%2810%29t+-+16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the second term on the right side 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+=+6+%2B+80t+-+16t%5E2 ALIGN=MIDDLE  ALT=\"h%28t%29+=+6+%2B+80t+-+16t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since we are looking at t = 1 second, substitute 1 for t and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%281%29+=+6+%2B+80+-+16 ALIGN=MIDDLE  ALT=\"h%281%29+=+6+%2B+80+-+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By combining the terms on the right side you confirm the answer that you were given, namely:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%281%29+=+70 ALIGN=MIDDLE  ALT=\"h%281%29+=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" > feet.<BR>\n" );
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document.write( "Hope this helps you. I still don't like this problem as a learning exercise in ballistic physics. </SPAN>\n" );
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