Math Problem Statement
Solution
To solve the problem of finding the dimensions of the rectangle inscribed in the ellipse with the given equation:
Steps:
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Understand the ellipse equation:
- The ellipse is centered at the origin with semi-major axis (horizontal) and semi-minor axis (vertical).
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Key idea for an inscribed rectangle:
- The vertices of the rectangle inscribed in the ellipse will lie on the ellipse.
- Let the vertices of the rectangle be , , , and .
- The rectangle's area is given by , and we want to find and under the ellipse's constraint.
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Ellipse equation constraint:
- The points must satisfy the ellipse equation:
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Optimize the area:
- Express in terms of using the ellipse equation:
- The rectangle's area is:
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Find that maximizes the area:
- Differentiate with respect to and set the derivative to zero: Simplify: Set : Multiply through by : Expand and solve:
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Find corresponding :
- Substitute into the ellipse equation:
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Dimensions of the rectangle:
- The rectangle has dimensions:
Final Answer:
The dimensions of the rectangle are:
Would you like a detailed explanation for any step?
Here are 5 related questions you may find helpful:
- What is the maximum area of the rectangle inscribed in the ellipse?
- How can Lagrange multipliers be applied to solve this problem?
- How do the dimensions of the rectangle relate to the ellipse’s semi-major and semi-minor axes?
- What happens if the ellipse is tilted (rotated)?
- Can this approach be generalized to ellipses with arbitrary semi-axis lengths?
Tip: Always consider the symmetry of ellipses when solving geometric problems—it simplifies calculations!
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Math Problem Analysis
Mathematical Concepts
Ellipse Geometry
Optimization
Area Maximization
Formulas
Equation of ellipse: (x^2/a^2) + (y^2/b^2) = 1
Area of rectangle: A = 4xy
Derivative of a function for optimization
Theorems
Symmetry of Ellipse
Optimization via Differentiation
Suitable Grade Level
Grades 11-12
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