document.write( "Question 1194693: A light at eye level stands 20 ft. from a house and 15 ft. from the path
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Algebra.Com's Answer #827030 by ikleyn(52790) $\"\"$ $\"About$

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document.write( "        +------------------------------------------------------+\r\n" );
document.write( "        |        The mathematical model is the TOP VIEW        |\r\n" );
document.write( "        |    from the bird flight height to the Earth surface. |\r\n" );
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document.write( "So, make a sketch following my wording description.\r\n" );
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document.write( "            You are a bird now, which flights very high \r\n" );
document.write( "            and looks at the Earth surface from the top.\r\n" );
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document.write( "You see the Earth surface as a plane; it would be better if you imagine the Earth surface \r\n" );
document.write( "as a coordinate plane.\r\n" );
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document.write( "Let the coordinate axis y=0 represents the wall of the house (x-axis).\r\n" );
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document.write( "The path leading from the house to the street is y-axis x=0; or more precisely, \r\n" );
document.write( "the positive ray along the y-axis\r\n" );
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document.write( "The light source is located at the point L = (15,20): 15 ft from the path (from y-axis)\r\n" );
document.write( "and 20 ft from the wall (from x-axis).\r\n" );
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document.write( "As a bird, you see the person from the top at the point P = (0,5): at the path, which is \r\n" );
document.write( "y-axis, and it gives you first coordinate x= 0; and 5 ft from the wall,\r\n" );
document.write( "which gives you the second coordinate y= 5.\r\n" );
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document.write( "Connect the point L (light source) and point P (the person) by a straight line\r\n" );
document.write( "and continue this line till its intersection with the wall (with x-axis).\r\n" );
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document.write( "This intersection represents the  SHADOW position  of the person at the wall.\r\n" );
document.write( "In this problem, your goal is to find x, based on the rest of data.\r\n" );
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document.write( "OK.  Now you can easily see in your sketch two similar right angled triangles.\r\n" );
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document.write( "One triangle has the legs 15 ft and 15 ft; the other triangle has the legs 5 ft and 5 ft.\r\n" );
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document.write( "So, the position of the shadow x at the time moment is x= 5 ft.\r\n" );
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document.write( "        But you should consider this picture in dynamics.\r\n" );
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document.write( "In dynamics, the leg of 15 ft, perpendicular to the wall, is changed in time as 15+6t feet,\r\n" );
document.write( "while the leg of 5 feet, perpendicular to the wall, is changed in time as 5-6t.\r\n" );
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document.write( "    Here, OBVIOUISLY, time t is counted starting from the moment \r\n" );
document.write( "              when the person is 5 ft from the wall.\r\n" );
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document.write( "Hence, from the triangles similarity, you can write this proportion\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "where x is your sought coordinate: the shadow position.\r\n" );
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document.write( "From the proportion, you get\r\n" );
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document.write( "    x =       (1)\r\n" );
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document.write( "Now all you need is to find the derivative x'(t) at the time moment t= 0.\r\n" );
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document.write( "This derivative is the speed of moving the shadow along the wall.\r\n" );
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document.write( "The rest is simple arithmetic, which I leave to you to complete the solution to the end.\r\n" );
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document.write( "Again, to complete, you should calculate the derivative x'(0), i.e. find x'(t)\r\n" );
document.write( "from equation (1) and substitute there t= 0.\r\n" );
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