document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196416>Question 1196416</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ball is thrown across a playing field from a height of 5 ft above the ground at an angle of 45 degrees to the horizontal at a speed of 20ft/s. It can be deduced from physical principles that the height of the ball is modeled by the function,<BR>\n" );
document.write( "h(x)=-32/(20)^2x^2+x+5 <BR>\n" );
document.write( "where x is the distance in feet that the ball has traveled horizontally.\r<BR>\n" );
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document.write( "(1) If the ball traveled horizontally 10 ft, what is the height?<BR>\n" );
document.write( "(2) Find the maximum height obtained by the ball. (enter two decimals points).<BR>\n" );
document.write( "(3) Find the horizontal distance the ball has traveled when it hits the ground. (enter two decimal points). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829276 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829276\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=-%2832%2F20%5E2%29x%5E2+%2Bx%2B5 ALIGN=MIDDLE  ALT=\"y+=-%2832%2F20%5E2%29x%5E2+%2Bx%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........simplify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=-%2832%2F400%29x%5E2+%2Bx%2B5 ALIGN=MIDDLE  ALT=\"y+=-%2832%2F400%29x%5E2+%2Bx%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=-%282%2F25%29x%5E2+%2Bx%2B5 ALIGN=MIDDLE  ALT=\"y+=-%282%2F25%29x%5E2+%2Bx%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(1) If the ball traveled horizontally 10ft, what is the height?\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=10 ALIGN=MIDDLE  ALT=\"x=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=-%282%2F25%2910%5E2+%2B10%2B5 ALIGN=MIDDLE  ALT=\"y+=-%282%2F25%2910%5E2+%2B10%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=7ft ALIGN=MIDDLE  ALT=\"y+=7ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If the ball traveled horizontally <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10ft ALIGN=MIDDLE  ALT=\"10ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the height is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7ft ALIGN=MIDDLE  ALT=\"7ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "(2)   Find the maximum height attained by the ball. (Round your answer to three decimal places.)\r<BR>\n" );
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document.write( "This is plainly a parabolic function - it fits the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since a is negative, it is an inverted parabola, one whose opening points downwards.  Such a parabola has a maximum value, which in this case corresponds to the maximum height reached by the ball. <BR>\n" );
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document.write( "The maximum or minimum value of a parabola occurs at the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=vertex ALIGN=MIDDLE  ALT=\"vertex\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which itself is located at the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coordinate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a ALIGN=MIDDLE  ALT=\"-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, based on our standard form.  \r<BR>\n" );
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document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=1+ ALIGN=MIDDLE  ALT=\"b+=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-%282%2F25%29 ALIGN=MIDDLE  ALT=\"a+=+-%282%2F25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F2a=-1%2F%282%28-2%2F25%29%29=25%2F4=6.25 ALIGN=MIDDLE  ALT=\"-b%2F2a=-1%2F%282%28-2%2F25%29%29=25%2F4=6.25\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " this corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6.25 ALIGN=MIDDLE  ALT=\"x+=+6.25\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We find the height the ball reaches at this point by plugging the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > into our equation.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-%282%2F25%29%2A%286.25%29%5E2+%2B+6.25+%2B+5+=+8.125ft ALIGN=MIDDLE  ALT=\"y+=+-%282%2F25%29%2A%286.25%29%5E2+%2B+6.25+%2B+5+=+8.125ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The maximum height attained by the ball is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.125ft ALIGN=MIDDLE  ALT=\"8.125ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "(3)\r<BR>\n" );
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document.write( "Find the horizontal distance the ball has traveled when it hits the ground. (Round your answer to one decimal place.)\r<BR>\n" );
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document.write( "The horizontal distance covered corresponds to the x value where the ball reaches a height of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  I.e., it's one of the roots of the function, which can be found with the quadratic equation, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Plugging in our coefficients, we have<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1+%2B-+sqrt%281%5E2-4%2A%28-%282%2F25%29%29%2A5%29%29%2F%282%28-%282%2F25%29%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+sqrt%281%5E2-4%2A%28-%282%2F25%29%29%2A5%29%29%2F%282%28-%282%2F25%29%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=x+=+%28-1+%2B-sqrt%281%2B8%2F5%29%29%2F%28-0.16%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-sqrt%281%2B8%2F5%29%29%2F%28-0.16%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=x+=+%28-1+%2B-+sqrt%282.6%29%29%2F%28-0.16%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+sqrt%282.6%29%29%2F%28-0.16%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=x=-6.25+%28-1+%2B-+1.61245%29 ALIGN=MIDDLE  ALT=\"x=-6.25+%28-1+%2B-+1.61245%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=x=+6.25+%2B-9.44 ALIGN=MIDDLE  ALT=\"x=+6.25+%2B-9.44\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There is one positive solution and one negative solution to this - for physical reasons, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=positive ALIGN=MIDDLE  ALT=\"positive\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+one ALIGN=MIDDLE  ALT=\"+one\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the one we prefer, since we're looking for where the ball hits the ground ahead of the thrower. <BR>\n" );
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document.write( "Thus, our answer is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+6.25+%2B9.44 ALIGN=MIDDLE  ALT=\"x=+6.25+%2B9.44\"   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=x+=+15.7ft ALIGN=MIDDLE  ALT=\"x+=+15.7ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the horizontal distance the ball has traveled when it hits the ground is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+15.7ft ALIGN=MIDDLE  ALT=\"+15.7ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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