Of all rectangles with a perimeter of 1313 , which one has the maximum area? (give the dimensions.) let a be the area of the rectangle. what is the objective function in terms of the width of the rectangle, w

Let the length be l.

Formula of perimeter is P = 2 (length + width)

1313 = 2 (l+w)

l = / frac{1313}{2}-w

And the formula of area of rectangle = length times width.

A = l*w

A = (/frac{1313}{2}-w) * w

A = / frac{1313w}{2} - w^2

And that's the required objective function.

The equation represents parabola and a parabola is maximum at its vertex.

And the formula of vertex is

w = - / frac{b}{2a} = -/frac{1313}{4}

Substituting this value of w in the formula of area, we will get

A = / frac{1313*1313}{8} - (/frac{1313}{4}) ^2

Area = / frac{1723969}{16}=107748 / square / units.