Question

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

Answer 1

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 .