document.write( "Question 575820: The sum of two positive numbers is 640, what should the numbers be if their product is as large as possible?\r
\n" ); document.write( "\n" ); document.write( "I really don't even know how to approach this problem. I started off by writing: x+2x=640. I don't know if that is the right way to approach the problem being that I keep getting it wrong when I try and submit the answer. Please help-Thank you :) \n" ); document.write( "
Algebra.Com's Answer #369655 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
It's sort of a trick problem.
\n" ); document.write( "If your numbers are x and y.
\n" ); document.write( "\"x%2By=640\" --> \"y=640-x\"
\n" ); document.write( "The product can be written as a function of x alone:
\n" ); document.write( "\"p%28x%29=x%28640-x%29=640x-x%5E2\"
\n" ); document.write( "You want to find the maximum value for that product.
\n" ); document.write( "\"p%28x%29\" is a quadratic function.
\n" ); document.write( "You may be studying quadratic functions, which graph as parabolas.
\n" ); document.write( "Do you know all about them and how to find the x for the maximum? If not, read on.
\n" ); document.write( "The book probably says that the general form is \"ax%5E2%2Bbx%2Bc\", and in your case a=-1, b=640 and c=0.
\n" ); document.write( "If the term in \"x%5E2\" has a positive coefficient (a>0), the parabola looks like a smile, opening up, and going through a minimum at its vertex, somewhere in the middle, like this: \"graph%28200%2C200%2C-6%2C14%2C-10%2C60%2C%28x-3%29%5E2-4%29\",
\n" ); document.write( "but a minus sign in front of the \"x%5E2\" (a<0) makes it frown like this \"graph%28200%2C200%2C-6%2C14%2C-10%2C60%2C-%28x-3%29%5E2%2B55%29\"
\n" ); document.write( "and then there is a maximum at the vertex.
\n" ); document.write( "Where is the maximum for \"p%28x%29=-x%5E2%2B640x\"?
\n" ); document.write( "If you are studying quadratic functions, you may be expected to invoke a formula that says that the axis of symmetry and x coordinate of the vertex/maximum is given by
\n" ); document.write( "\"x=-b%2F2a\" where b is the coefficient of x and a is the coefficient of \"x%5E2\".
\n" ); document.write( "In this case it would be \"x=-640-%282%2A%28-1%29%29=-640%2F%28-2%29=320\"
\n" ); document.write( "Otherwise, you may be expected to transform the function like this:
\n" ); document.write( "
\n" ); document.write( "and then say that since the square will be positive or zero, and it has that minus sign in front, the function will be \"320%5E2\" when xthe parenthesis is zero (x=320), and it will be less for any other x value.
\n" ); document.write( "Either way, you show that to make the product greatest one of the numbers has to be \"highlight%28320%29\",
\n" ); document.write( "and since they add up to 640, the other number is 320 too.
\n" ); document.write( "\"y=640-x\", so \"y=640-320=320\"
\n" ); document.write( "When the problem said \"two positive numbers\" you probably assumed they were two different numbers. The problem did not say they were different. That's the trick.
\n" ); document.write( "If two numbers have to add up to some constant, their product will be maximum when they are the same number.
\n" ); document.write( "Similar problems talk about the perimeter of a rectangle, which is the sum of length plus width, doubled. If the perimeter of a rectangle has to be a certain number, the area will be maximum when length and width are the same, meaning that the rectangle is a square. \n" ); document.write( "
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