(Solved): Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area th ...

Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse \( \frac{x^{2}}{9}+\frac{y^{2}}{25}=1 \) with sides parallel to the coordinate axes. Let length be the dimension parallel to the \( x \)-axis and let width be the dimension parallel to the \( y \)-axis. Length \( = \) Width \( = \) (Type exact answers, using radicals as needed.)