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?

Verified step by step guidance

1. Consider the semicircle with radius 5 centered at the origin on the coordinate plane. The equation of the semicircle is given by \( y = \sqrt{25 - x^2} \).
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 \((x, \sqrt{25 - x^2})\).
3. 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 \(A = 2x \cdot \sqrt{25 - x^2}\).
4. 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.
5. 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.

Key Concepts

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.