document.write( "Question 1174226: A ladder 4m long at a construction site is resting against a wall. The bottom of the ladder is slipping away from the wall. Find the estimate of the instantaneous rate of change of the Height H of the top of the ladder with respect to the Distance D of the bottom of the ladder from the wall when the bottom of the ladder is 2.5m away from the wall. Use h = 0.01 as the central interval. \n" ); document.write( "

Algebra.Com's Answer #851952 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A ladder 4m long at a construction site is resting against a wall. The bottom of the ladder is slipping away
\n" ); document.write( "from the wall. Find the estimate of the instantaneous rate of change of the Height H of the top of the ladder
\n" ); document.write( "with respect to the Distance D of the bottom of the ladder from the wall
\n" ); document.write( "when the bottom of the ladder is 2.5m away from the wall. Use h = 0.01 as the central interval.
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\n" ); document.write( "\n" ); document.write( "First, the problem in the post is posed INCORRECTLY.
\n" ); document.write( "To be correct, the problem must provide the rate of the horizontal move of the base of the ladder
\n" ); document.write( "from the wall, as the input data. Without this input data, the problem CAN NOT BE SOLVED.\r
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\n" ); document.write( "\n" ); document.write( "Second, this instruction \"Use h = 0.01 as the central interval\" is IRRELEVANT to the problem.
\n" ); document.write( "By knowing the mathematical level of those who write to this forum, I am 837% sure
\n" ); document.write( "that this \"instruction\" came mistakenly from the other problem.\r
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\n" ); document.write( "\n" ); document.write( "Third, the \"solution\" by @CPhill is non-sensical collection/soup of words.
\n" ); document.write( "It does not solve the problem, and even can not be a subject for discussions.
\n" ); document.write( "So, my advise to you is to ignore his post, for the sake of safety of your mind.\r
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\n" ); document.write( "\n" ); document.write( "Now I am ready to start my solution.
\n" ); document.write( "I will assume, that the rate of the horizontal move of the base of the ladder
\n" ); document.write( "from the wall is 0.5 meters per second.\r
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document.write( "                           S O L U T I O N\r\n" );
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document.write( "Let H be the height of the ladder (in meters) when it leans against the wall.\r\n" );
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document.write( "Let D be the distance of the ladder base from the wall (in meters).\r\n" );
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document.write( "Write the Pythagorean equation \r\n" );
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document.write( "    H^2 + D^2 = 4^2 = 16,    (1)\r\n" );
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document.write( "where '4' is the ladder's length, in meters.\r\n" );
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document.write( "When the ladder moves, we consider H and D as functions of time;  H = H(t),  D = D(t).\r\n" );
document.write( "But the length of the ladder remains with no change.\r\n" );
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document.write( "Differentiate equation (1) with respect to time, keeping in mind that the right side is a constant.\r\n" );
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document.write( "You will get\r\n" );
document.write( "                                                        -D*D'\r\n" );
document.write( "    2H*H' + 2D*D' = 0,  or  H*H' + D*D' = 0,  or  H' = -------.    (2)\r\n" );
document.write( "                                                          H\r\n" );
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document.write( "Calculate H:  H =  = 3.1225 m  (approximately).\r\n" );
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document.write( "Recall that D' is given  D' = 0.5 m/s.\r\n" );
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document.write( "Substitute the values into formula (2) and get\r\n" );
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document.write( "    H' =  = -0.40032 m/s.\r\n" );
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document.write( "ANSWER.  Thus the ladder's upper point moves down along the vertical wall \r\n" );
document.write( "         with the rate of 0.4 m/s at the assigned time moment.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved correctly, with complete explanations.\r
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\n" ); document.write( "\n" ); document.write( "For this class of problems, it is a standard methodology of their solution.\r
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