Problem 104 - Algebra and Trigonometry

Question

A rectangle is to be inscribed in a semicircle of radius 5 \(\mathrm{cm}\) as shown in the following figure. (a) Show that the area of the rectangle is modeled by the function $$A(\theta)=25 \sin 2 \theta$$ (b) Find the largest possible area for such an inscribed rectangle. (c) Find the dimensions of the inscribed rectangle with the largest possible area.

Step-by-Step Solution

Answer

(a) Verified: \(A(\theta) = 25 \sin 2\theta\), (b) Max area is 25 cm\(^2\), (c) Dimensions: 5\sqrt{2} cm \( \times \) \(\frac{5\sqrt{2}}{2}\) cm.

Step 1: Understand the Geometry

Visualize the semicircle with a rectangle inscribed in it. The semicircle has a radius of 5 cm, centered at the origin, with its flat side along the x-axis. The rectangle is symmetric about the y-axis, with its top corners touching the circle.

Step 2: Set Up the Coordinates

Position the top vertices of the rectangle at points \((x, y)\) and \((-x, y)\). Since the rectangle is inscribed in the semicircle, the equation of the semicircle is \(x^2 + y^2 = 25\) (because the radius is 5 cm).

Step 3: Express y in Terms of x

Rewrite the semicircle equation in terms of \(y\), such that \(y = \sqrt{25 - x^2}\).

Step 4: Relate x and y with \(\theta)

Let \( \theta \) be the angle that the line connecting the origin to the point \((x, y)\) makes with the positive x-axis. Therefore, \(x = 5 \cos \theta\) and \(y = 5 \sin \theta\).

Step 5: Model the Area of the Rectangle

The width of the rectangle is \(2x = 10 \cos \theta\) and the height is \(y = 5 \sin \theta\). Therefore, the area \(A\) of the rectangle is given by \(A = 2x \times y = 10 \cos \theta \times 5 \sin \theta = 50 \cos \theta \sin \theta\). Use the identity \(\sin 2\theta = 2 \sin \theta \cos \theta\) to express \(A\) as \(A(\theta) = 25 \sin 2\theta\).

Step 6: Find the Maximum Area

The area function \(A(\theta) = 25 \sin 2\theta\) is maximized when \(\sin 2\theta = 1\). This occurs when \(2\theta = \frac{\pi}{2}\), and thus \(\theta = \frac{\pi}{4}\). Substitute back to find the maximum area: \(A = 25 \times 1 = 25 \text{ cm}^2\).

Step 7: Determine Dimensions for Maximum Area

For \(\theta = \frac{\pi}{4}\), calculate \(x\) and \(y\) as: \(x = 5 \cos \frac{\pi}{4} = \frac{5\sqrt{2}}{2}\) and \(y = 5 \sin \frac{\pi}{4} = \frac{5\sqrt{2}}{2}\). Thus, width = 2x = \(5\sqrt{2}\) cm and height = \(\frac{5\sqrt{2}}{2}\) cm.

Key Concepts

Area of Rectangle

Understanding how the area of a rectangle is calculated is crucial when analyzing geometric problems. For a rectangle, the area is obtained by multiplying its width by its height. In the context of a rectangle inscribed in a semicircle, this concept becomes particularly interesting.

Given the semicircle's properties, we recognize the width of the rectangle as twice the x-coordinate of the top right corner, \(2x\), and the height as the y-coordinate, \(y\). This is because the rectangle is symmetric about the y-axis. Thus, the area of the rectangle can be expressed as follows:

• Width: \(2x = 10 \cos \theta\)
• Height: \(y = 5 \sin \theta\)
• Area: \(A = 2x \times y = 10 \cos \theta \times 5 \sin \theta = 50 \cos \theta \sin \theta\)

Utilizing trigonometric identities simplifies these expressions. For instance, the identity \(\sin 2\theta = 2 \sin \theta \cos \theta\) allows us to represent the area function as: \(A(\theta) = 25 \sin 2\theta\). This format reveals the relationship between the angle \(\theta\) and the area, enabling further analysis to find the optimal rectangle configuration.

Trigonometric Functions

Trigonometric functions play a pivotal role in understanding this geometric setup. These functions help express coordinate relationships in a semicircle, given angles relative to the x-axis.

Key trigonometric functions in this problem include sine and cosine. They are used to relate the coordinates \(x\) and \(y\) of a point on the semicircle to the angle \(\theta\):

• \(x = 5 \cos \theta\)
• \(y = 5 \sin \theta\)

These equations stem from the definitions of sine and cosine in a right triangle, where the radius (hypotenuse) is 5, and the sides are \(x\) and \(y\). As such, these functions allow us to parameterize the rectangle’s position within the semicircle.

Moreover, using trigonometric identities like \(\sin 2\theta = 2 \sin \theta \cos \theta\) not only simplifies the expressions but also proves instrumental in deriving the area function. By converting product terms into identities, deeper insights into maximum and minimum values are attained.

Optimization Problem

Optimization involves finding the best possible solution— in this case, the rectangle with the largest area. Optimization problems in mathematics often require setting up an equation that is dependent on a variable, and then finding the maximum or minimum value this equation can achieve.

For the inscribed rectangle problem, the task is to maximize the area function \(A(\theta) = 25 \sin 2\theta\). This necessitates understanding when \(\sin 2\theta\) reaches its peak, as the other constants don't change the function's behavior regarding optimization.

The function \(\sin \theta\) or \(\sin 2\theta\) reaches its maximum at values where \(\theta = \frac{\pi}{4}\). At this point, \(\sin 2\theta = 1\), leading directly to the maximum possible area, which is 25 cm².

Subsequently, identifying \(\theta\) allows calculation of the dimensions:

• \(x = 5 \cos \frac{\pi}{4} = \frac{5\sqrt{2}}{2}\)
• \(y = 5 \sin \frac{\pi}{4} = \frac{5\sqrt{2}}{2}\)

These values ensure the configuration that yields the largest area. Thus, the optimization process not only highlights the maximum value but also provides the precise geometric arrangement to achieve it.