Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# area of the largest rectangle?

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• Pope
Lv 7
1 month ago

The ellipse has horizontal major radius 5 and vertical minor radius 3. Its center is the origin.

Begin with the circle of radius 5, centered on the origin. The largest inscribed rectangle is a square of diagonal length 10.

area of square = (10)²/2 = 50

Let the square be oriented so that its sides are parallel to the coordinate axes. Now, scale the figure vertically by factor 3/5, with the x-axis invariant. This transformation maps the circle to the given ellipse, and the square to the inscribed rectangle of greatest area. The transformation scales all areas by the same factor.

rectangle area = (3/5)(50) = 30

• rotchm
Lv 7
1 month ago

Consider the graph in a cartesian plane. Let (x,y) be one of the vertex of the rectangle which lies on the ellipse.

The area is A = 2x * 2y.

But y = 3√(1 - x²/25).  [why?].

Thus a = 2x*2*3√(1 - x²/25) = 12x√(1 - x²/25).

Unanon yourself and we will take it from there.