Rectangles beneath a semicircle A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

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Consider the semicircle with radius 5 centered at the origin on the coordinate plane. The equation of the semicircle is given by \( y = \sqrt{25 - x^2} \).

Let the base of the rectangle be along the x-axis, with vertices at \((-x, 0)\) and \((x, 0)\). The other two vertices will be \((-x, \sqrt{25 - x^2})\) and \((x, \sqrt{25 - x^2})\).

The width of the rectangle is \(2x\) and the height is \(\sqrt{25 - x^2}\). Therefore, the area \(A\) of the rectangle can be expressed as \(A = 2x \cdot \sqrt{25 - x^2}\).

To find the maximum area, we need to differentiate the area function \(A(x) = 2x \cdot \sqrt{25 - x^2}\) with respect to \(x\) and set the derivative equal to zero to find critical points.

Solve the equation \(\frac{dA}{dx} = 0\) to find the value of \(x\) that maximizes the area. Then, substitute this value back into the expressions for the width and height to find the dimensions of the rectangle.

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주요 개념

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Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to determine the dimensions of the rectangle that maximize its area, which requires setting up a function for the area in terms of the rectangle's dimensions and then using techniques such as differentiation to find critical points.

Area of a Rectangle

The area of a rectangle is calculated by multiplying its length by its width. In this context, the rectangle's dimensions are constrained by the semicircle, so we express the area as a function of one variable, typically the width, and derive the corresponding height using the semicircle's equation.

Semicircle Equation

The equation of a semicircle can be expressed as y = √(r² - x²), where r is the radius. For a semicircle with a radius of 5, the equation becomes y = √(25 - x²). This relationship is crucial for determining the height of the rectangle based on its base's position along the diameter, allowing us to express the area function accurately.