document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195307>Question 1195307</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A line was found to be 1500ft was laid out using a 50-ft steel tape under a pull of 12lbs at a mean temperature of 20 degree Celsius. Determine the correct length of the line if the tape used is of standard length at 25°C under a pull of 10 lbs. Assume the cross-sectional area of tape to be 0.003 sq.in, 0.0000116 ft/ °C per as its coefficient of thermal expansion, and 30 x 10° psi as its elastic modulus. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848448 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848448\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ## Determining the Correct Length of the Line\r<BR>\n" );
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document.write( "We can solve this problem by considering the following factors that affect the length of the steel tape:\r<BR>\n" );
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document.write( "1. **Thermal Expansion:** As temperature increases, the steel tape expands.<BR>\n" );
document.write( "2. **Tension:** Pulling the tape stretches it slightly.\r<BR>\n" );
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document.write( "We can use the following steps to find the correct length of the line:\r<BR>\n" );
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document.write( "**1. Calculate the Change in Length due to Thermal Expansion:**\r<BR>\n" );
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document.write( "* **Formula:** ΔL_temp = α * L₀ * ΔT<BR>\n" );
document.write( "    * ΔL_temp: Change in length due to temperature (°ft)<BR>\n" );
document.write( "    * α: Coefficient of thermal expansion (°ft/°C)<BR>\n" );
document.write( "    * L₀: Original length of the tape (ft)<BR>\n" );
document.write( "    * ΔT: Change in temperature (°C)\r<BR>\n" );
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document.write( "* **Values:**<BR>\n" );
document.write( "    * α = 0.0000116 ft/°C<BR>\n" );
document.write( "    * L₀ = 50 ft (assuming the tape is standard length)<BR>\n" );
document.write( "    * ΔT = 25°C - 20°C = 5°C\r<BR>\n" );
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document.write( "* **Calculation:**<BR>\n" );
document.write( "    * ΔL_temp = 0.0000116 ft/°C * 50 ft * 5°C<BR>\n" );
document.write( "    * ΔL_temp = 0.0029 ft ≈ 0.003 ft (round to 3 decimal places)\r<BR>\n" );
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document.write( "**2. Calculate the Change in Length due to Tension:**\r<BR>\n" );
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document.write( "* We need additional information about the relationship between tension and length change for this specific steel tape. \r<BR>\n" );
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document.write( "**Here, we can make two assumptions:**\r<BR>\n" );
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document.write( "**Assumption 1: Linear Relationship**\r<BR>\n" );
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document.write( "If the relationship between tension (T) and change in length (ΔL_tension) is linear, we can use the following formula:\r<BR>\n" );
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document.write( "* ΔL_tension = k * (T - T₀)<BR>\n" );
document.write( "    * k: proportionality constant (ft/lb) representing the change in length per unit change in tension<BR>\n" );
document.write( "    * T: Applied tension (lb)<BR>\n" );
document.write( "    * T₀: Reference tension (lb)\r<BR>\n" );
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document.write( "We don't have the value of k for this specific tape. However, we can acknowledge that a higher tension (12 lbs) will cause a greater change in length compared to the reference tension (assumed to be 10 lbs).\r<BR>\n" );
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document.write( "**Assumption 2: Information from Manufacturer**\r<BR>\n" );
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document.write( "The manufacturer might provide a table or graph showing the change in length for different tension values. This information can be used to determine ΔL_tension.\r<BR>\n" );
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document.write( "**3. Apply Corrections to Find the Correct Length:**\r<BR>\n" );
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document.write( "* Since the thermal expansion increases the length and the tension might decrease the length (depending on the tension-length relationship), we can assume a net **decrease** in length.\r<BR>\n" );
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document.write( "* **Correct Length (L_correct):**<BR>\n" );
document.write( "    * L_correct = L₀ - ΔL_temp - ΔL_tension\r<BR>\n" );
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document.write( "* **Note:** Due to the lack of information about ΔL_tension, we cannot calculate the exact correct length.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* Based on the available information, we can determine that the steel tape likely experienced a slight **decrease** in length due to thermal expansion (0.003 ft).<BR>\n" );
document.write( "* The tension might have caused a further decrease in length, but the magnitude is unknown without additional information.\r<BR>\n" );
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document.write( "Therefore, the actual length of the line (L_correct) is likely **shorter** than the measured length (1500 ft) by an unknown amount.<BR>\n" );
document.write( " </SPAN>\n" );
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