Question

Maximum perimeter rectangle Use Lagrange multipliers to find the dimensions of the rectangle with the maximum perimeter that can be inscribed with sides parallel to the coordinate axes in the ellipse x2/a2 + y2/b2 = 1.

Answer 1

The perimeter function of a rectangle is given as follows,

Equation of the ellipse is given by,

So, the functionwill be,

Now, findandas below:

Now, consider the Lagrange’s multiplier is, so,

…… (1)

Solve,

Put them into the constraint expression and get:

Solve and get:

Now, from the expressions (1), one can get the sets:

When,

When,

Negative value is not possible for side length. So, the maximum side lengths will be:

So, the maximum dimensions will be.