document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1113459>Question 1113459</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A wire 10 cm long is cut into two pieces, one of length x and the other of length x and the other of length 10-x. Each piece is bent into a shape of square. Which of the following is the model for the total area enclosed by the two squares of function of x?<BR>\n" );
document.write( "a. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=+%28x%5E2-5x%2B50%29%2F8 ALIGN=MIDDLE  ALT=\"A%28x%29=+%28x%5E2-5x%2B50%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=+%282x%5E2%2B100%29%2F16 ALIGN=MIDDLE  ALT=\"A%28x%29=+%282x%5E2%2B100%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "c. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=+%28x%5E2-10x%2B50%29%2F8 ALIGN=MIDDLE  ALT=\"A%28x%29=+%28x%5E2-10x%2B50%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "d. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%29=+%28x%5E2%2B100%29%2F16 ALIGN=MIDDLE  ALT=\"A%28x%29=+%28x%5E2%2B100%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "After finding the total area, what is the value of x that minimizes the total area of the two squares? CHOICES(3 cm, 4cm, 5cm, and 6cm) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #728492 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=728492\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The first (x) length makes a square that has area <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%5B1%5D+=++%28x%2F4%29%5E2+ ALIGN=MIDDLE  ALT=\"+A%5B1%5D+=++%28x%2F4%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The 2nd length is 10-x, so it makes a square that has area <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%5B2%5D+=+%28%2810-x%29%2F4%29%5E2+ ALIGN=MIDDLE  ALT=\"+A%5B2%5D+=+%28%2810-x%29%2F4%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A%5B1%5D+%2B+A%5B2%5D+=++%28x%2F4%29%5E2+%2B+%28%2810-x%29%2F4%29%5E2+=++%28x%5E2+%2B+%28100-20x%2Bx%5E2%29%29%2F16++=+highlight%28+%28x%5E2+-+10x+%2B+50%29%2F8+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I will use Calculus to find the minimum:\r<BR>\n" );
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document.write( "First we take the 1st derivative of A with respect to x (giving us the rate of change of A with respect to x):<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28dA%2Fdx%29+=+%282x-10%29%2F8+ ALIGN=MIDDLE  ALT=\"+%28dA%2Fdx%29+=+%282x-10%29%2F8+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  —>  This is zero when x = 5cm.   So 5cm is a critical point.\r<BR>\n" );
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document.write( "The 2nd derivative of A with respect to x tells us if the function is curved up or curved down (at x=5):<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28d%5E2A%2Fdx%5E2%29+=+2+ ALIGN=MIDDLE  ALT=\"+%28d%5E2A%2Fdx%5E2%29+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >       so the curve is curved up (concave up) everywhere which means the critical point is at a minimum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+5cm+%29+ ALIGN=MIDDLE  ALT=\"+highlight%28+5cm+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > minimizes the area. <BR>\n" );
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document.write( "Since this problem was multiple choice, the endpoints x=0 and x=10  need not be considered.   In solving min/max problems in general, one must consider the function value at the endpoints in addition to at all critical points found.  <BR>\n" );
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document.write( "To figure out the minimum without Calculus, one way to see the minimum is to plot the graph of the function (x along the x-axis,  A(x) along the y axis).  Since this was multiple choice, you'd plot (compute A(x)) for the 4 values given, and compare their heights.  When plotting a function in general you would take use x values and plot more A(x) values.  <BR>\n" );
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