To find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with side length \( L \), where one side of the rectangle lies on the base of the triangle, we start by establishing the configuration.
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Triangle Basics: An equilateral triangle can be oriented such that one side (the base) lies along the x-axis, with its vertices at \( (0, 0) \), \( (L, 0) \), and \( \left( \frac{L}{2}, \frac{L \sqrt{3}}{2} \right) \).
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Rectangle Dimensions: Let the rectangle have width \( w \) and height \( h \). The width of the rectangle will be \( w \) and will extend symmetrically from the center, which means the left corner is at \( \frac{L}{2} - \frac{w}{2} \) and the right corner is at \( \frac{L}{2} + \frac{w}{2} \).
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Rectangle Height: The height of the rectangle, \( h \), can be calculated based on the position of the top corners of the rectangle within the triangle. The highest point of the triangle is at \( \left( \frac{L}{2}, \frac{L \sqrt{3}}{2} \right) \).
Using similar triangles, the height \( h \) of the rectangle at any point on the base can be expressed in terms of \( w \):
- The height of the triangle decreases linearly as we move horizontally. The slope of the triangle sides can be determined. The line from \( (0,0) \) to \( \left( \frac{L}{2}, \frac{L \sqrt{3}}{2} \right) \) has the equation: \[ y = \sqrt{3} x \]
- The height \( h \) at \( x = \frac{L}{2} \pm \frac{w}{2} \): \[ h = \sqrt{3} \left( \frac{L}{2} - \frac{w}{2} \right) \]
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Area of the Rectangle: The area \( A \) of the rectangle can thus be expressed as: \[ A = w \cdot h = w \cdot \sqrt{3} \left( \frac{L}{2} - \frac{w}{2} \right) \] Which simplifies to: \[ A = \frac{\sqrt{3}}{2} w \left( L - w \right) \]
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Maximizing the Area: We want to maximize the function: \[ A(w) = \frac{\sqrt{3}}{2} w (L - w) \] To find the maximum, we take the derivative of \( A \) with respect to \( w \) and set it to zero: \[ \frac{dA}{dw} = \frac{\sqrt{3}}{2} (L - 2w) \] Setting this equal to zero gives: \[ L - 2w = 0 \implies w = \frac{L}{2} \]
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Finding height: Substituting \( w = \frac{L}{2} \) back to find \( h \): \[ h = \sqrt{3} \left( \frac{L}{2} - \frac{L}{4} \right) = \sqrt{3} \cdot \frac{L}{4} = \frac{L \sqrt{3}}{4} \]
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Final Dimensions: Thus, the dimensions of the rectangle of largest area inscribed in the triangle are:
- Width \( w = \frac{L}{2} \)
- Height \( h = \frac{L \sqrt{3}}{4} \)
The rectangle of largest area that can be inscribed in the equilateral triangle has dimensions: \[ \text{Width: } \frac{L}{2}, \quad \text{Height: } \frac{L \sqrt{3}}{4} \]
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