document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181807>Question 1181807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The volume of the rectangular block in the figure is calculated from the following measurements of its dimensions:  �10.00±0.10 cm,   �5.00±0.06 cm   and   �4.00±0.04 cm. Calculate the error in the value of the volume of the block assuming:  a) the errors are independent b) the errors are correlated such that they all push the estimate in the same direction. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849963 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=849963\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the error in the volume, considering both independent and correlated errors:\r<BR>\n" );
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document.write( "**1. Calculate the Nominal Volume:**\r<BR>\n" );
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document.write( "V = lwh = (10.00 cm)(5.00 cm)(4.00 cm) = 200.00 cm³\r<BR>\n" );
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document.write( "**2.  Fractional Uncertainties:**\r<BR>\n" );
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document.write( "*   Δl/l = 0.10 cm / 10.00 cm = 0.01<BR>\n" );
document.write( "*   Δw/w = 0.06 cm / 5.00 cm = 0.012<BR>\n" );
document.write( "*   Δh/h = 0.04 cm / 4.00 cm = 0.01\r<BR>\n" );
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document.write( "**a) Independent Errors:**\r<BR>\n" );
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document.write( "When errors are independent, we add the *squares* of the fractional uncertainties and then take the square root.\r<BR>\n" );
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document.write( "1.  Fractional uncertainty in volume:<BR>\n" );
document.write( "    (ΔV/V)² = (Δl/l)² + (Δw/w)² + (Δh/h)²<BR>\n" );
document.write( "    (ΔV/V)² = (0.01)² + (0.012)² + (0.01)²<BR>\n" );
document.write( "    (ΔV/V)² = 0.0001 + 0.000144 + 0.0001 = 0.000344<BR>\n" );
document.write( "    ΔV/V = √0.000344 ≈ 0.0185\r<BR>\n" );
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document.write( "2.  Absolute uncertainty in volume:<BR>\n" );
document.write( "    ΔV = (ΔV/V) * V = 0.0185 * 200.00 cm³ ≈ 3.7 cm³\r<BR>\n" );
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document.write( "3.  Volume with uncertainty:<BR>\n" );
document.write( "    V = 200.0 ± 3.7 cm³\r<BR>\n" );
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document.write( "**b) Correlated Errors:**\r<BR>\n" );
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document.write( "When errors are fully correlated (all pushing the estimate in the same direction), we simply *add* the fractional uncertainties.\r<BR>\n" );
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document.write( "1.  Fractional uncertainty in volume:<BR>\n" );
document.write( "    ΔV/V = Δl/l + Δw/w + Δh/h<BR>\n" );
document.write( "    ΔV/V = 0.01 + 0.012 + 0.01 = 0.022\r<BR>\n" );
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document.write( "2.  Absolute uncertainty in volume:<BR>\n" );
document.write( "    ΔV = (ΔV/V) * V = 0.022 * 200.00 cm³ = 4.4 cm³\r<BR>\n" );
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document.write( "3.  Volume with uncertainty:<BR>\n" );
document.write( "    V = 200.0 ± 4.4 cm³\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "*   **Independent Errors:** V = 200.0 ± 3.7 cm³<BR>\n" );
document.write( "*   **Correlated Errors:** V = 200.0 ± 4.4 cm³\r<BR>\n" );
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document.write( "As expected, the uncertainty is larger when the errors are correlated because they all contribute to the error in the same direction. When independent, there is a chance that some errors cancel each other out, thus reducing the total uncertainty.<BR>\n" );
document.write( " </SPAN>\n" );
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