document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161857>Question 1161857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a fifteen long piece of wire is cut and formed into a circle and an equilateral triangle. graph the function for total area of the shapes.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #785492 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=785492\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Assume the 15 unit piece of wire is cut at C units from one end where C is the circumference of the circle that is formed from that piece.  That leaves 15 - C units with which to form the equilateral triangle.  Since the perimeter of an equilateral triangle is 3s where s is the measure of one side of the triangle, we know that 3s = 15 - C, which is to say\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  s\ =\ \frac{15\,-\,C}{3}\">\r<BR>\n" );
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document.write( "The radius of a circle as a function of the circumference is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  r\ =\ \frac{C}{2\pi}\">\r<BR>\n" );
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document.write( "The area of an equilateral triangle in terms of the measure of a side is derived from the facts that an altitude to one side is a perpendicular bisector of that side and it bisects the apex angle so that the altitude is the long leg of a 30-60-90 right triangle.  Since the hypotenuse measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s\">, the short leg is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{s}{2}\">, and the long leg measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{s\sqrt{3}}{2}\">.  The area of the 30-60-90 triangle is the measure of the short leg times the measure of the long leg divided by 2.  But since the whole equilateral triangle is two of these right triangles, the area of the whole equilateral triangle is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{s^2\sqrt{3}}{4}\">\r<BR>\n" );
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document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s\ =\ \frac{15\ -\ C}{3}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(15\,-\,C)^2\sqrt{3}}{36}\">\r<BR>\n" );
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document.write( "Since the area of a circle is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \pi{r^2}\">\r<BR>\n" );
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document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\ =\ \frac{C}{2\pi}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \pi{\(\frac{C}{2\pi}\)^2}\ =\ \frac{C^2}{4\pi}\">\r<BR>\n" );
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document.write( "Then the total area of the two shapes as a function of the circumference of the circle shape is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ A_{total}(C)\ =\ \frac{C^2}{4\pi}\ +\  \frac{(15\,-\,C)^2\sqrt{3}}{36}\">\r<BR>\n" );
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document.write( "The graph is a parabola, but I'll let you work out the details.<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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