document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1127011>Question 1127011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> hello, I have this problem that I need to explained to my daughter. please help.\r<BR>\n" );
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document.write( "\"A rocket is launched and follows a parabolic reaching a maximum height of 14 km when is is 8 km away from launch pad, which 250 m above the ground. Determine the height of rocket as a function of its horizontal distance from lunch pad.\"\r<BR>\n" );
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document.write( "thank you 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 #743356 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=743356\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "A rocket is launched and follows a parabolic arch reaching a max height of 14 km when it is 8 km away from the launch pad, which is 250 m above the ground. Determine the height of the rocket as a function of its horizontal distance from the launch pad<BR>\n" );
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document.write( "Let<BR>\n" );
document.write( "x = horizontal distance in meters<BR>\n" );
document.write( "y = vertical distance (or height) in meters\r<BR>\n" );
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document.write( "There are three points on this graph<BR>\n" );
document.write( "A = (0,250)<BR>\n" );
document.write( "B = (8000,14000)<BR>\n" );
document.write( "C = (16000,250)\r<BR>\n" );
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document.write( "Point A is the launch point at 250 m above the ground, which is why we have y = 250. I made x = 0 as the starting x value, though technically you can start with any x value you want. The next point, point B, is the vertex point. This point is 8 km or 8000 meters away from the launch point which explains why x = 8000. The y coordinate is 14000 meters (14 km) so y = 14000. The third point C is another 8000 meters from point B, but in the opposite direction as point A, and it will have the same height as point A because the parabola has symmetry along the vertical line that runs through the vertex.\r<BR>\n" );
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document.write( "The equation y = ax^2 + bx + c is the general form of any parabola. We need to find the values of a,b,c which will help us find the function we want.\r<BR>\n" );
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document.write( "Plug in (x,y) = (0,250) to find the value of c<BR>\n" );
document.write( "y = ax^2 + bx + c <BR>\n" );
document.write( "ax^2 + bx + c = y<BR>\n" );
document.write( "a(0)^2 + b(0) + c = 250<BR>\n" );
document.write( "0a + 0b + c = 250<BR>\n" );
document.write( "0 + 0 + c = 250<BR>\n" );
document.write( "c = 250\r<BR>\n" );
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document.write( "Now plug in (x,y) = (8000,14000)<BR>\n" );
document.write( "y = ax^2 + bx + c <BR>\n" );
document.write( "ax^2 + bx + c = y<BR>\n" );
document.write( "a(8000)^2 + b(8000) + 250 = 14000<BR>\n" );
document.write( "64000000a + 8000b + 250 = 14000<BR>\n" );
document.write( "64000000a + 8000b  + 250-250 = 14000-250<BR>\n" );
document.write( "64000000a + 8000b = 13750\r<BR>\n" );
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document.write( "Solve the previous equation for b<BR>\n" );
document.write( "64000000a + 8000b = 13750<BR>\n" );
document.write( "8000b = 13750-64000000a<BR>\n" );
document.write( "8000b = -64000000a+13750<BR>\n" );
document.write( "b = (-64000000a+13750)/8000<BR>\n" );
document.write( "b = (-64000000a)/8000+(13750)/8000<BR>\n" );
document.write( "b = -8000a+1.71875<BR>\n" );
document.write( "we'll plug this equation in later\r<BR>\n" );
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document.write( "Go back to the general form of a parabola. Plug in (x,y) = (16000,250)<BR>\n" );
document.write( "y = ax^2 + bx + c <BR>\n" );
document.write( "ax^2 + bx + c = y<BR>\n" );
document.write( "a(16000)^2 + b(16000) + 250 = 250<BR>\n" );
document.write( "256000000a + 16000b + 250 = 250<BR>\n" );
document.write( "256000000a + 16000b + 250-250 = 250-250<BR>\n" );
document.write( "256000000a + 16000b = 0\r<BR>\n" );
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document.write( "Now replace b with -8000a+1.71875 and solve for 'a'<BR>\n" );
document.write( "256000000a + 16000b = 0<BR>\n" );
document.write( "256000000a + 16000( b ) = 0<BR>\n" );
document.write( "256000000a + 16000(-8000a+1.71875) = 0<BR>\n" );
document.write( "256000000a - 128000000a + 27500 = 0<BR>\n" );
document.write( "128,000,000a + 27500 = 0<BR>\n" );
document.write( "128,000,000a = -27500<BR>\n" );
document.write( "a = -27500/(128,000,000)<BR>\n" );
document.write( "a = -0.00021484375\r<BR>\n" );
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document.write( "Use this value of 'a' to find b<BR>\n" );
document.write( "b = -8000a+1.71875<BR>\n" );
document.write( "b = -8000(-0.00021484375)+1.71875<BR>\n" );
document.write( "b = 3.4375\r<BR>\n" );
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document.write( "We have enough info to go from this<BR>\n" );
document.write( "y = ax^2 + bx + c<BR>\n" );
document.write( "to this<BR>\n" );
document.write( "y = -0.00021484375x^2 + 3.4375x + 250\r<BR>\n" );
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document.write( "The function is therefore:<BR>\n" );
document.write( "f(x) = -0.00021484375x^2 + 3.4375x + 250<BR>\n" );
document.write( "x is the horizontal distance and f(x) is the height.\r<BR>\n" );
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document.write( "If you were to plug in x = 0, then you should get f(x) = 250. <BR>\n" );
document.write( "If you were to plug in x = 8000, then you should get f(x) = 14000.<BR>\n" );
document.write( "If you were to plug in x = 16000, then you should get f(x) = 250.<BR>\n" );
document.write( "These three facts will confirm that we have the proper parabolic function.<BR>\n" );
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