Question

A rectangle has two corners on the \(x\) -axis and the other two on the curve \(y=\cos x\), with \(-\pi / 2

Short answer

The dimensions are approximately \(1.4796\times 0.7398\).

Step-by-step solution

Understanding the Problem

We need to find the dimensions of a rectangle that is bounded by the curve on the top and the x-axis, with the maximum possible area. The rectangle's top corners lie on the curve \( y = \cos x \) and the bottom corners are on the x-axis within \( -\frac{\pi}{2} < x < \frac{\pi}{2} \).

Identify Expression for Rectangle's Area

The area \( A \) of the rectangle can be expressed as \( A = 2x \cdot y \), where \( 2x \) is the width (the rectangle spans from \(-x\) to \(x\)) and \( y \) is the height given by the curve, \( y = \cos x \). Therefore, the area function is \( A(x) = 2x \cos x \).

Derive the Area Function

To optimize the area, we need to find the derivative of the area function \( A(x) = 2x \cos x \). Using the product rule, \( A'(x) = 2 \cos x - 2x \sin x \).

Find Critical Points

Set the derivative \( A'(x) = 0 \) to find critical points: \( 2\cos x - 2x \sin x = 0 \). Simplifying, we get \( \cos x = x \sin x \). This equation must be solved on the interval \( -\frac{\pi}{2} < x < \frac{\pi}{2} \).

Solve for Critical Points Numerically

Solving \( \cos x = x \sin x \) analytically is challenging, so use numerical methods or graphing to find approximate solutions. The solution for the maximum area where \( x \) is approximately \( 0.7398 \).

Calculate Maximum Area and Dimension

Substitute \( x = 0.7398 \) into the equations for dimensions: Width is \( 2x = 1.4796 \), and height is \( y = \cos 0.7398 \approx 0.7398 \). Thus, the dimensions are approximately \( 1.4796 \times 0.7398 \).

Key concepts

Critical Points

In calculus, finding the critical points of a function is crucial for optimization problems. These are the points where the derivative of the function equals zero or does not exist. Finding critical points helps us determine where the function could reach its maximum or minimum values.

For the rectangle bounded by the curve and the x-axis, we need to identify critical points of the area function, which is derived from the given conditions of the rectangle's dimensions. This involves setting the derivative, in this case, of the area function, equal to zero and solving for the variable. Critical points serve as candidate solutions for locating potential maxima or minima of the function, which determines the optimal solution of our problem - the maximum area of the rectangle.

Area Maximization

Area maximization involves finding the dimensions of a geometric shape that yield the largest possible area. In this problem, we are maximizing the area of a rectangle situated between the x-axis and the cosine curve.

For a rectangle with one side on the x-axis and vertices on the curve, the area depends on the product of its width and height. We are given that the width is double a horizontal distance, represented as \(2x\), and the height is the corresponding \(y\)-value from the curve, \(y = \cos x\). The goal is to choose a value of \(x\) that maximizes the resulting area \(A(x) = 2x \cos x\). This requires analyzing the behavior of the area function within the specified interval using calculus.

Derivative

The derivative is a powerful tool in calculus that measures how a function changes as its input changes. It helps in determining the slope of a function at any point and is used extensively in optimization problems to find maxima and minima.

In this problem, we find the derivative of the area function \(A(x) = 2x \cos x\) to locate critical points. We use the product rule, which allows us to differentiate products of two functions, resulting in \(A'(x) = 2 \cos x - 2x \sin x\). By setting this derivative to zero, we identify the points where the slope of the area function is zero, indicating potential maximum or minimum points for \(A(x)\). Thus, derivatives play a central role in determining the optimal dimensions of the rectangle for maximum area.

Numerical Methods

Numerical methods are techniques to approximate solutions for mathematical problems that are difficult or impossible to solve analytically. They are especially useful when dealing with complex functions or equations.

In this exercise, the equation \(\cos x = x \sin x\) was difficult to solve by algebraic methods alone; hence, numerical methods or graphing were employed to find an approximate solution. These methods allow for finding solutions to optimization problems, such as the dimensions granting maximum area, when the analytical approach becomes overly complex. Approximations obtained through numerical methods often guide us in making informed choices in practical applications.