document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203907>Question 1203907</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a norman windo with a perimeter of 20 units has the shape of rectangle surmounted by a semi-circle. If its base is x units long and the height of the rectangle is y units long, write a mathematical model for its area in terms of x.<BR>\n" );
document.write( "  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839806 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839806\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you want to find the area in terms of x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "perimeter of the circle is equal to 2 * p * x<BR>\n" );
document.write( "x is the radius of the the circle<BR>\n" );
document.write( "perimeter of the semi-circle is half that = p * x \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "perimeter of the rectangle is equal to 2x + 2y <BR>\n" );
document.write( "2x is one length of the rectangle.<BR>\n" );
document.write( "y is one width of the rectangle.<BR>\n" );
document.write( "only one length of the rectangle is used because the other length is the base of the semi-circle which is internal to the structure and therefore not counted as part of the perimeter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the perimeter of the window is the perimeter of the semi-circle plus the perimeter of the rectangle.<BR>\n" );
document.write( "since the perimeter of the window is equal to 20, you get:<BR>\n" );
document.write( "20 = pi * x + 2x + 2y<BR>\n" );
document.write( "solve for 2y to get:<BR>\n" );
document.write( "2y = 20 - pi * x - 2x<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = 10 - pi/2 * x - x *****\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area of the circle is equal to pi * x^2<BR>\n" );
document.write( "the area of the semi-circle is equal to pi * x^2 / 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area of the rectangle is equal to 2x * y\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area of the window is equal to the area of the semi-circle plus the area of the rectangle which is equal to pi * x^2/2 + 2x * y *****\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you have y = 10 - pi/2 * x - x<BR>\n" );
document.write( "replace y with that in the formula for the area of the window to get:<BR>\n" );
document.write( "area of the window = pi * x^2/2 + 2x * (10 - pi/2 * x - x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplify this to get:<BR>\n" );
document.write( "area of the window = pi * x^2/2 + 20x - pi * x^2 - 2x^2<BR>\n" );
document.write( "that should be your solution.<BR>\n" );
document.write( "it is the area of the window in terms of x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to see if this makes sense, find a value for x that makes the value of y positive.<BR>\n" );
document.write( "i chose 2.<BR>\n" );
document.write( "it's chosen randomly and tested to make sure the value of y is positive when using it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when x = 2, the value of y = 10 - pi/2 * x - x which becomes equal to 10 - pi/2 * 2 - 2 which becomes equal to 4.858407346.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when x = 2 and y = 4.858407346, you get:<BR>\n" );
document.write( "area of the window = pi * x^2/2 + 2x * y which becomes equal to pi * 2^2/2 + 2 * 2 * 4.858407346 which is equal to 25.71681469\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to get the formula for the area of the window in in terms of x, replace y with (10 - pi/2 * x - x) to get:<BR>\n" );
document.write( "the area of the window = pi * x^2/2 + 2x * (10 - pi/2 * x - x) which becomes equal to pi * 2^2/2 + 2*2 * (10 - pi/2 * 2 - 2) which is equal to 25.71681469.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this confirms thqt the formula for the area of the window in terms of x is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula is area of the window = pi * x^2/2 + 2x * (10 - pi/2 * x - x) .<BR>\n" );
document.write( "this can be simplified to:<BR>\n" );
document.write( "area of the window = 20x - 2x^2 - pi/2 * x^2<BR>\n" );
document.write( "if you combine like terms, the formula becomes:<BR>\n" );
document.write( "area of the window = 20x - (2 + pi/2) * x^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i checked the numbers several times and i think i finaly got it down right.<BR>\n" );
document.write( "the answer should be:<BR>\n" );
document.write( "area of the window equals:<BR>\n" );
document.write( "20x - (2 + pi/2) * x^2 or:<BR>\n" );
document.write( "20x - 2x^2 - pi/2 * x^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );