document.write( "Question 399513: why does a square 45x45 have more sqaure footage than a rectangle 30x60. both could be made from the same two 90' pieces \n" ); document.write( "
Algebra.Com's Answer #282909 by Alan3354(69443)![]() ![]() You can put this solution on YOUR website! The simplest answer is that 45*45 is more than 30*60. \n" ); document.write( "For a given perimeter (180 would make a 45 by 45, not 90) the max area of a rectangle is a square. \n" ); document.write( "------------ \n" ); document.write( "For a perimeter of 180 \n" ); document.write( "P = 2W + 2L = 180 \n" ); document.write( "w + L = 90 \n" ); document.write( "Area = W*L = W*(90 - W) \n" ); document.write( "To find the max area as a function of W: \n" ); document.write( "f(W) = w(90 - W) = 90W - W^2 \n" ); document.write( "90W - W^2 - area = 0 \n" ); document.write( "-W^2 + 90W - area = 0 \n" ); document.write( "This is a parabola that is open downward. \n" ); document.write( "The line of symmetry is W = -b/2a = -90/-2 = 45 \n" ); document.write( "The max is on the line of symmetry: \n" ); document.write( "f(45) = 90*45 - 45^2 = 4050 - 2025 \n" ); document.write( "Max area = 2025 sq units \n" ); document.write( "W = 45 \n" ); document.write( "L = 90-45 = 45 \n" ); document.write( "------------ \n" ); document.write( "Any other pair that add to 90 will have a smaller product. \n" ); document.write( "It will be (45 + x)*(45 - x) = 2025 - x^2 \n" ); document.write( "For any x > 0, it will be less than 2025. \n" ); document.write( "-------------------- \n" ); document.write( "Note: 2025 is the max area of a rectangle with a perimeter of 180. \n" ); document.write( "A circle will have the max area of all figures. \n" ); document.write( " \n" ); document.write( " |