document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180642>Question 1180642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h =  -  0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet.   A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h =  -  0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet.   \r<BR>\n" );
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document.write( "a.Create a own quadratic relation that would model the path of a shot from a distance of 15 feet that would hit the rim of the basketball net. Explain how you obtained the answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850097 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850097\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to create a quadratic relation that models a basketball shot hitting the rim from 15 feet away:\r<BR>\n" );
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document.write( "**Understanding the Problem:**\r<BR>\n" );
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document.write( "* We're given the standard quadratic model for a basketball shot: h = -0.2d² + 3d + 6<BR>\n" );
document.write( "    * h: height of the ball<BR>\n" );
document.write( "    * d: distance from the shooter\r<BR>\n" );
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document.write( "* We need to create a new quadratic model where:<BR>\n" );
document.write( "    * The shot is taken from 15 feet away (d = 15).<BR>\n" );
document.write( "    * The ball hits the rim (we'll assume a standard rim height of 10 feet, so h = 10 when d = 15).\r<BR>\n" );
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document.write( "**Creating the New Quadratic Model:**\r<BR>\n" );
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document.write( "1. **General Form:** Start with the general form of a quadratic relation: h = ad² + bd + c\r<BR>\n" );
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document.write( "2. **Use the Given Information:**<BR>\n" );
document.write( "    * We know the shot is from 15 feet and hits the 10-foot rim, so we have a point (d, h) = (15, 10).  Substitute this into the equation:<BR>\n" );
document.write( "        10 = a(15)² + b(15) + c<BR>\n" );
document.write( "        10 = 225a + 15b + c\r<BR>\n" );
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document.write( "    * To make the shot hit the rim at this distance, we need the vertex of the parabola to be at d = 15. The x-coordinate (in this case, d-coordinate) of the vertex of the parabola is given by -b/2a.  So:<BR>\n" );
document.write( "        15 = -b / 2a<BR>\n" );
document.write( "        -30a = b\r<BR>\n" );
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document.write( "3. **Choose a Value for 'a':**<BR>\n" );
document.write( "    * We have some freedom here. Let's choose a value for 'a' that's different from the original model but still makes sense for a basketball shot.  A slightly smaller value for 'a' would make the shot arc higher. Let's try a = -0.15.\r<BR>\n" );
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document.write( "4. **Solve for 'b' and 'c':**<BR>\n" );
document.write( "    * Using -30a = b and a = -0.15, we get:<BR>\n" );
document.write( "        b = -30 * (-0.15) = 4.5\r<BR>\n" );
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document.write( "    * Substitute a = -0.15 and b = 4.5 into the equation 10 = 225a + 15b + c:<BR>\n" );
document.write( "        10 = 225(-0.15) + 15(4.5) + c<BR>\n" );
document.write( "        10 = -33.75 + 67.5 + c<BR>\n" );
document.write( "        c = -23.75\r<BR>\n" );
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document.write( "**The New Quadratic Model:**\r<BR>\n" );
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document.write( "* h = -0.15d² + 4.5d - 23.75\r<BR>\n" );
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document.write( "This model represents a basketball shot taken from 15 feet away that hits the rim of the basketball net.\r<BR>\n" );
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document.write( "**Explanation:**\r<BR>\n" );
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document.write( "* We used the given information (distance and height of the rim) to create an equation with the general form of a quadratic relation.<BR>\n" );
document.write( "* We used the fact that the vertex of the parabola should be at the rim (d = 15) to relate 'a' and 'b'.<BR>\n" );
document.write( "* By choosing a value for 'a', we could solve for 'b' and 'c' to complete the model. </SPAN>\n" );
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