document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174226>Question 1174226</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850704 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=850704\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Set up the Problem**\r<BR>\n" );
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document.write( "* **Ladder Length:** 4 meters (constant)<BR>\n" );
document.write( "* **Distance from Wall (D):** 2.5 meters<BR>\n" );
document.write( "* **Height on Wall (H):** We need to find this.<BR>\n" );
document.write( "* **Rate of Change:** We want to find dH/dD (how the height changes with respect to the distance).\r<BR>\n" );
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document.write( "**2. Use the Pythagorean Theorem**\r<BR>\n" );
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document.write( "* We have a right triangle formed by the wall, the ground, and the ladder.<BR>\n" );
document.write( "* The Pythagorean theorem states: D² + H² = 4² (where 4 is the ladder length)<BR>\n" );
document.write( "* D² + H² = 16\r<BR>\n" );
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document.write( "**3. Express Height (H) in Terms of Distance (D)**\r<BR>\n" );
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document.write( "* H² = 16 - D²<BR>\n" );
document.write( "* H = √(16 - D²)\r<BR>\n" );
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document.write( "**4. Find the Instantaneous Rate of Change (dH/dD)**\r<BR>\n" );
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document.write( "* We'll use the given central difference method to estimate the derivative.<BR>\n" );
document.write( "* The formula for the central difference is:<BR>\n" );
document.write( "    * dH/dD ≈ [H(D + h) - H(D - h)] / (2h)<BR>\n" );
document.write( "* Where:<BR>\n" );
document.write( "    * D = 2.5 meters<BR>\n" );
document.write( "    * h = 0.01 meters\r<BR>\n" );
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document.write( "**5. Calculate H(D + h) and H(D - h)**\r<BR>\n" );
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document.write( "* H(D + h) = H(2.5 + 0.01) = H(2.51) = √(16 - 2.51²) ≈ √(16 - 6.3001) ≈ √9.6999 ≈ 3.11446<BR>\n" );
document.write( "* H(D - h) = H(2.5 - 0.01) = H(2.49) = √(16 - 2.49²) ≈ √(16 - 6.2001) ≈ √9.7999 ≈ 3.13048\r<BR>\n" );
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document.write( "**6. Substitute into the Central Difference Formula**\r<BR>\n" );
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document.write( "* dH/dD ≈ (3.11446 - 3.13048) / (2 * 0.01)<BR>\n" );
document.write( "* dH/dD ≈ (-0.01602) / 0.02<BR>\n" );
document.write( "* dH/dD ≈ -0.801\r<BR>\n" );
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document.write( "**Result**\r<BR>\n" );
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document.write( "* The estimated instantaneous rate of change of the height of the ladder with respect to the distance of the bottom of the ladder from the wall is approximately -0.801 meters per meter.\r<BR>\n" );
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document.write( "**Therefore, when the bottom of the ladder is 2.5 meters from the wall, the top of the ladder is sliding down the wall at a rate of approximately 0.801 meters for every meter the bottom slides away.**<BR>\n" );
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