Final answer:
To find the dimensions of the rectangle with the largest perimeter inside an ellipse using Lagrange multipliers, set up the objective function for perimeter, apply the constraint of the ellipse equation, and solve the equations derived from setting the gradient of perimeter to lambda times the gradient of the constraint.
Step-by-step explanation:
The student is asking how to find the dimensions of the rectangle with the largest perimeter that can be inscribed inside an ellipse, where the sides of the rectangle are parallel to the coordinate axes. To solve this, we employ the Lagrange multiplier technique. Let the coordinates of the vertices of the rectangle be at (+/- x, +/- y). Since the ellipse is centered at the origin and the rectangle is symmetric about the axes, we only need to consider one quadrant.
First, set up the objective function for the perimeter, which is P = 2x + 2y (since in one quadrant, the vertices are at (x,0) and (0,y)). Next, we need to apply the constraint given by the ellipse equation. For the ellipse x^2/81 + y^2/144 = 1, the constraint g(x, y) = x^2/81 + y^2/144 - 1 = 0. Using Lagrange multipliers, we set the gradient of P equal to lambda times the gradient of g, which gives two equations: 2 = lambda*(2x/81) and 2 = lambda*(2y/144).
By solving these equations simultaneously and substituting back into the constraint equation of the ellipse, we can find the optimal values for x and y, which gives us the dimensions of the rectangle with the largest perimeter.