document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=844186>Question 844186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cannon ball is fired almost vertically upwards from ground level. The cannon ball has height given by the relationship H = 12t – t2 metres, where t is the time in seconds after firing.\r<BR>\n" );
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document.write( "Consider the following:\r<BR>\n" );
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document.write( "(i) If we sketch a graph of the height H against the time t after firing what shape will result?<BR>\n" );
document.write( "•	Draw an accurate sketch of this graph and label the graph in the context of this scenario.\r<BR>\n" );
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document.write( "(ii) How long would it take for the cannonball to reach its maximum height? <BR>\n" );
document.write( "•	What would be the maximum height reached? <BR>\n" );
document.write( "•	Re-write the relationship H in a different form using the values you obtained for the maximum height?\r<BR>\n" );
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document.write( "(iii) The maximum height reached by the cannonball is doubled.  Given that the cannonball will still land after the same flight time write down a relationship for H, which would represent a doubling of the height.\r<BR>\n" );
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document.write( "(iv) Investigate further variations in the height reached. Place this data in a table and reflect on your results.\r<BR>\n" );
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document.write( "(v) Investigate what would happen if the original height is maintained but the flight time is varied.\r<BR>\n" );
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document.write( "I need help understanding the math side of this, thanks for your help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #508749 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=508749\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A cannon ball is fired almost vertically upwards from ground level.<BR>\n" );
document.write( " The cannon ball has height given by the relationship H = 12t – t^2 metres, where t is the time in seconds after firing.<BR>\n" );
document.write( "Consider the following:<BR>\n" );
document.write( "(i) If we sketch a graph of the height H against the time t after firing what shape will result?<BR>\n" );
document.write( "• Draw an accurate sketch of this graph and label the graph in the context of this scenario.<BR>\n" );
document.write( "I suggest you plot these points, replace x, find y<BR>\n" );
document.write( " x | y<BR>\n" );
document.write( "-------<BR>\n" );
document.write( " 0 | 0<BR>\n" );
document.write( " 2 | 20<BR>\n" );
document.write( " 4 | 32<BR>\n" );
document.write( " 6 | 36<BR>\n" );
document.write( " 8 | 32<BR>\n" );
document.write( " 9 | 27<BR>\n" );
document.write( "12 | 0 (ground level)<BR>\n" );
document.write( "Look something like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-4%2C+15%2C+-10%2C+100%2C+-x%5E2%2B12x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-4%2C+15%2C+-10%2C+100%2C+-x%5E2%2B12x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( ":<BR>\n" );
document.write( "(ii) How long would it take for the cannonball to reach its maximum height?<BR>\n" );
document.write( "• What would be the maximum height reached?<BR>\n" );
document.write( "You can see it reaches max height when t = 6 sec (halfway between 0 and 12<BR>\n" );
document.write( "• Re-write the relationship H in a different form using the values you obtained for the maximum height?<BR>\n" );
document.write( "In the equation h = 12t - t^2, replace t with 6<BR>\n" );
document.write( "h = 12(6) - 6^2<BR>\n" );
document.write( "h = 72 - 36<BR>\n" );
document.write( "h = 36m (as is shown on the graph)<BR>\n" );
document.write( ":<BR>\n" );
document.write( "(iii) The maximum height reached by the cannonball is doubled.<BR>\n" );
document.write( " Given that the cannonball will still land after the same flight time,<BR>\n" );
document.write( " write down a relationship for H, which would represent a doubling of the height.<BR>\n" );
document.write( "Using the form y = ax^2 + bx + c, we have to find new values for a & b, (no c)<BR>\n" );
document.write( "For the coordinates:<BR>\n" );
document.write( " x = 6, y = 72 (twice the height)<BR>\n" );
document.write( "and<BR>\n" );
document.write( "x = 12, y = 0 (same flight time)<BR>\n" );
document.write( "write an equation for each pair replace x and y<BR>\n" );
document.write( "36a + 6b = 72<BR>\n" );
document.write( "and<BR>\n" );
document.write( "144a + 12b = 0<BR>\n" );
document.write( "multiply the 1st equation by 2, subtract from the 2nd equation<BR>\n" );
document.write( "144a + 12b = 0<BR>\n" );
document.write( "72a + 12b = 144<BR>\n" );
document.write( "-----------------subtraction eliminates b, find a<BR>\n" );
document.write( "72a = -144<BR>\n" );
document.write( "a = -144/72<BR>\n" );
document.write( "a = -2<BR>\n" );
document.write( "Find b using the 1st original equation<BR>\n" );
document.write( "36(-2) + 6b = 72<BR>\n" );
document.write( "-72 + 6b = 72<BR>\n" );
document.write( "6b = 72 + 72<BR>\n" );
document.write( "6b = 144<BR>\n" );
document.write( " b = 144/6<BR>\n" );
document.write( " b = 24\r<BR>\n" );
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document.write( "Graph the new equation h = 24t - 2t^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-4%2C+15%2C+-10%2C+100%2C+-2x%5E2%2B24x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-4%2C+15%2C+-10%2C+100%2C+-2x%5E2%2B24x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "NOte that it reaches 72m in 6 sec now <BR>\n" );
document.write( ":<BR>\n" );
document.write( "(iv) Investigate further variations in the height reached. Place this data in a table and reflect on your results.<BR>\n" );
document.write( "Note that if you want to double the height and retain the same time<BR>\n" );
document.write( "double the coefficients of t and t^2, for example to double the height again to<BR>\n" );
document.write( "144m<BR>\n" );
document.write( "48t - 4t^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+15%2C+-50%2C+150%2C+-4x%5E2%2B48x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+15%2C+-50%2C+150%2C+-4x%5E2%2B48x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can make the table here using this example<BR>\n" );
document.write( ": <BR>\n" );
document.write( "(v) Investigate what would happen if the original height is maintained but the flight time is varied.<BR>\n" );
document.write( "Using the form ax^2 + bx = y, find the a and b of the new equation<BR>\n" );
document.write( "x=10, y=36; max occurs in 10 sec, but is still 36<BR>\n" );
document.write( "x=20, y=0; hits the ground in 20 sec<BR>\n" );
document.write( "100a + 10b = 36<BR>\n" );
document.write( "400a + 20b = 0<BR>\n" );
document.write( "Mult the 1st equation by 2, subtract from the 2nd<BR>\n" );
document.write( "400a + 20b = 0<BR>\n" );
document.write( "200a + 20b = 72<BR>\n" );
document.write( "----------------subtraction eliminates b, find a<BR>\n" );
document.write( "200a = -72<BR>\n" );
document.write( "a = -72/200<BR>\n" );
document.write( "a = -.36<BR>\n" );
document.write( "Find b using the 1st original equation<BR>\n" );
document.write( "100(-.36) + 10b = 36<BR>\n" );
document.write( "-36 + 10b = 36<BR>\n" );
document.write( "10b = 72<BR>\n" );
document.write( "b = 7.2<BR>\n" );
document.write( "the new equation: h = 7.2t - .36t^2, this graph looks like <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-10%2C+24%2C+-10%2C+50%2C+-.36x%5E2%2B7.2x%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-10%2C+24%2C+-10%2C+50%2C+-.36x%5E2%2B7.2x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "max occurs at 10 sec but is still 36 m  </SPAN>\n" );
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