Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.
d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to 90° .)

Verified step by step guidance

1. Consider an arbitrary triangle with a given area A. We want to inscribe a rectangle of maximum area within this triangle.
2. Position the triangle in the coordinate plane such that one of its sides lies along the x-axis. This simplifies the problem by allowing us to use the base and height of the triangle in our calculations.
3. Let the base of the triangle be b and the height be h. The area of the triangle is given by A = (1/2) * b * h.
4. To maximize the area of the inscribed rectangle, we can use calculus. Let the width of the rectangle be x and the height be y. The area of the rectangle is A_rect = x * y.
5. Using similar triangles, express y in terms of x, b, and h. Then, substitute this expression into A_rect to get a function of x. Differentiate this function with respect to x, set the derivative to zero, and solve for x to find the dimensions that maximize the area of the rectangle.

Key Concepts

Inscribed Figures

An inscribed figure is a shape placed within another shape such that all vertices of the inscribed figure touch the boundary of the outer shape. In this context, we are interested in a rectangle inscribed within a triangle, which means that the rectangle's corners will touch the sides of the triangle. Understanding how to position the rectangle optimally is crucial for maximizing its area.

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 yield the maximum area while being constrained by the triangle's sides. This typically involves using techniques such as derivatives to find critical points and applying the second derivative test to confirm whether these points yield a maximum area.

Area of a Triangle

The area of a triangle can be calculated using the formula A = 1/2 * base * height. This concept is essential because the area of the triangle provides a constraint for the dimensions of the inscribed rectangle. Knowing the area allows us to relate the dimensions of the rectangle to the triangle's geometry, which is vital for solving the problem effectively.