document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1081529>Question 1081529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Kirby moves with constant speed 5 units per hour along the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%2F4+%2B6 ALIGN=MIDDLE  ALT=\"y+=+3x%2F4+%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > crossing<BR>\n" );
document.write( "the y-axis at midnight and the x-axis later. When is the x-axis crossing made? What does it<BR>\n" );
document.write( "mean to say that Kirby’s position is a function of time? What is Kirby’s position 1.5 hours<BR>\n" );
document.write( "after midnight? What is Kirby’s position t hours after midnight? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #695621 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=695621\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> kirby moves at a constant speed of 5 units per hour along the line y = 3x/4 + 6\r<BR>\n" );
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document.write( "the equation of the line can be written as y = 3/4 * x + 6\r<BR>\n" );
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document.write( "this is the general form of y = mx + b\r<BR>\n" );
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document.write( "the slope is 3/4 and the y-intercept is 6.\r<BR>\n" );
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document.write( "the convention is that the slope measures the movement along the line going from left to right.\r<BR>\n" );
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document.write( "if you are moving along the line from right to left, then the movement is reversed.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "assuming your slope is 3/4 and you are moving along the line from the left side of the graph to the right side of the graph, which is the norm, then <BR>\n" );
document.write( "you move up 3 units along the y-axis for every 4 units you move to the right along the x-axis.\r<BR>\n" );
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document.write( "however, assuming your slope is 3/4 and you are moving along the line from the right side of the graph to the left side of the graph, then you move down 3 units along the y-axis for every 4 units you move to the left along the x-axis.\r<BR>\n" );
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document.write( "your equation is y = 3/4 * x + 6\r<BR>\n" );
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document.write( "to find your y-intercept, make the value of x = 0 and solve for y.\r<BR>\n" );
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document.write( "you will get y = 3/4 * 0 + 6 which results in y = 6.\r<BR>\n" );
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document.write( "your y-intercept is at y = 6 which is the coordinate point of (0,6).\r<BR>\n" );
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document.write( "to find your x-intercept, make the value of y = 0 and solve for x.\r<BR>\n" );
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document.write( "you will get 0 = 3/4 * x + 6.\r<BR>\n" );
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document.write( "when you solve for x, you will find that your x-intercept is at x = -8 which is the coordinate point of (-8,0).\r<BR>\n" );
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document.write( "the following graph of your equation shows that this is true.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/051701.jpg\" alt=\"$$$\"\r<BR>\n" );
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document.write( "for every hour, you move 5 units along the line.\r<BR>\n" );
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document.write( "since your slope is 3/4, this means that, for every hour, you move 3/5 * 5 units = 3 units along the y-axis and 4/5 * 5 = 4 units along the x-axis.\r<BR>\n" );
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document.write( "if you are moving from left to right on the graph, then you are moving up 3 units up and 4 units to the right from your previous position.\r<BR>\n" );
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document.write( "if you are moving from right to left on the graph, then you are moving down 3 units and 4 units to the left from your previous position.\r<BR>\n" );
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document.write( "based on the picture of the graph i showed you above, you will be moving from the right side of the graph to the left side of the graph along the line.\r<BR>\n" );
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document.write( "that means, for every hour, you move 5 units along the line from the rightmost point along the line to the leftmost point along the line.\r<BR>\n" );
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document.write( "that also means, for every hour, you move 3 units down along the y-axis and 4 units to the left along the x-axis.\r<BR>\n" );
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document.write( "if you look at the y-intercept point and the x-intercept point, you will find that you are going from (0,6) to (-8,0)\r<BR>\n" );
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document.write( "the distance you travel along the line is the square root of (the square of the change in your x-coordinate value plus the square of the change in your y-coordinate value).\r<BR>\n" );
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document.write( "that becomes the square root of (64 + 36) = square root of 100 = 10.\r<BR>\n" );
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document.write( "the distance along the line from your y-intercept to your x-intercept is 10 units.\r<BR>\n" );
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document.write( "since you move 5 units per hour along the line, that means it takes 2 hours to get from the y-intercept to the x-intercept.\r<BR>\n" );
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document.write( "in 2 hours, your downward movement along the y-axis is 3 * 2 = 6\r<BR>\n" );
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document.write( "in 2 hours, your movement to the left along the x-axis is 4 * 2 = 8.\r<BR>\n" );
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document.write( "since you started from (0,6), you would wind up at (-8,0).\r<BR>\n" );
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document.write( "your movement along the line is 5 units per hour going from the right side of the graph to the left.\r<BR>\n" );
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document.write( "your movement along the y-axis is be 3 units down every hour.\r<BR>\n" );
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document.write( "your movement along along the x-axis is be 4 units to the left every hour.\r<BR>\n" );
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document.write( "if you allow the number of hours you travel to be equal to t, then your formula would be:\r<BR>\n" );
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document.write( "distance along the line is 5 * t moving from the right side of the graph to the left side of the graph.\r<BR>\n" );
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document.write( "distance along the y-axis would be 3 * t moving down along the y-axis.\r<BR>\n" );
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document.write( "distance along the x-axis would be 4 * t moving to the left along the x-axis.\r<BR>\n" );
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document.write( "you would use this formula as follows:\r<BR>\n" );
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document.write( "assume 2 hours travel time.\r<BR>\n" );
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document.write( "distance along the line is 10 units when you travel from the right side of the graph to the left side of the graph.\r<BR>\n" );
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document.write( "distance down along the y-axis is 3 * 2 = 6 units down.\r<BR>\n" );
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document.write( "distance to the left along the x-axis is 4 * 2 = 8 units to the left.\r<BR>\n" );
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document.write( "the new point is (0,6) minus (8,6) = (-8,0) which we already know is correct, based on what we see on the first graph shown.\r<BR>\n" );
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document.write( "if she travels for 1.5 hours from midnight, then:\r<BR>\n" );
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document.write( "the distance along the line would  be 7.5 units traveling from the right side of the graph to the left side.\r<BR>\n" );
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document.write( "distance down along the y-axis would be 3 * 1.5 = 4.5 units.\r<BR>\n" );
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document.write( "distance to the left along the x-axis would be 4 * 1.5 = 6 units.\r<BR>\n" );
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document.write( "the new position would be (0,6) minus (6,4.5) = (-6,1.5)\r<BR>\n" );
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document.write( "this new position can be seen on the following graph.\r<BR>\n" );
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document.write( "your distance traveled in a downward direction is (6-1.5) = 4.5 units<BR>\n" );
document.write( "your distance traveled in a right to left direction is (0-(-6)) = 6 units.<BR>\n" );
document.write( "your distance traveled along the line is the hypotenuse of the right triangle formed which is equal to square root of (4.5^2 + 6^2) which is equal to square root of (20.25 + 36) which is equal to square root of 56.25 which is equal to 7.5.\r<BR>\n" );
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document.write( "everything checks out so we can derive the final formula you need as follows:\r<BR>\n" );
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document.write( "the x-coordinate of kirby's position t hours after midnight would be 0 minus 4 * t<BR>\n" );
document.write( "the y-coordinate of kirby's position t hours after midnight would be 6 minus 3 * t<BR>\n" );
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document.write( "when t = 1.5 after midnight, you get:\r<BR>\n" );
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document.write( "x-coordinate of the new position is 0 - (4 * 1.5) = 0 - 6 = -6.\r<BR>\n" );
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document.write( "y-coordinate of the new position is 6 - (3 * 1.5) = 6 - 4/5 = 1.5.\r<BR>\n" );
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document.write( "the coordinate of the new position is (-6,1.5)\r<BR>\n" );
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document.write( "this agrees with what is shown on the second graph above.\r<BR>\n" );
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document.write( "the general formula works for any number of hours.\r<BR>\n" );
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document.write( "movement along the line is in a right side of the graph to the left side of the graph direction.\r<BR>\n" );
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document.write( "since the line is sloping up in a left side of the graph to the right side of the graph direction, then the line is sloping down in a right side of the graph to the left side of the graph direction.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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