document.write( "Question 1200217: A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=9−x2. What are the dimensions of such a rectangle with the greatest possible area? \n" ); document.write( "

Algebra.Com's Answer #834278 by ikleyn(52874) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "A rectangle is inscribed with its base on the x-axis and its upper corners
\n" ); document.write( "on the parabola y=9−x2. What are the dimensions of such a rectangle
\n" ); document.write( "with the greatest possible area?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The area of any such inscribed rectangle is \r\n" );
document.write( "\r\n" );
document.write( "    A = A(x) = base*height = (2x)*y = 2x*(9-x^2) = 18x - 2x^3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We want maximize the area A(x), i.e. maximize function A(x).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we take the derivative and equate it to zero\r\n" );
document.write( "\r\n" );
document.write( "    A'(x) = 18 - 6x^2 = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From this equation, we find\r\n" );
document.write( "\r\n" );
document.write( "    6x^2 = 18  --->  x^2 = 18/6 = 3  --->  x = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The dimensions of the inscribed rectangle are    horizontally\r\n" );
document.write( "and   = 9 - 3 = 6 units vertically.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The maximum area is  xy =  =   square units.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );