Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the shape of the triangle of largest area that can be inscribed in a circle of radius \(r\). Hint: Let \(\alpha, \beta\), and \(\gamma\) be the central angles that subtend the three sides of the triangle. Show that the area of the triangle is \(\frac{1}{2} r^{2}[\sin \alpha+\sin \beta-\sin (\alpha+\beta)]\). Maximize.

Step-by-Step Solution

Answer

An equilateral triangle inscribed in the circle provides the largest area.

Step 1: Understand the Problem

We need to find the triangle with the largest possible area that can be inscribed in a circle with radius \( r \). This means we have to look for the configuration of the triangle that maximizes its area.

Step 2: Explore the Triangle Inscribed in a Circle

An inscribed triangle in a circle can be defined by three central angles \( \alpha, \beta \), and \( \gamma \) such that \( \alpha + \beta + \gamma = 2\pi \). These angles subtend the sides of the triangle on the circle.

Step 3: Express Area in terms of Central Angles

The area \( A \) of a triangle inscribed in a circle with angles \( \alpha \), \( \beta \), and \( \gamma \) can be expressed using the sine rule. The area of the triangle is given by: \[ A = \frac{1}{2} r^{2} (\sin \alpha + \sin \beta + \sin \gamma) \]Using \( \gamma = 2\pi - (\alpha + \beta) \), we substitute \( \sin \gamma = \sin(2\pi - (\alpha + \beta)) = -\sin(\alpha + \beta) \), leading to the area formula:\[ A = \frac{1}{2} r^{2} [\sin \alpha + \sin \beta - \sin(\alpha + \beta)] \]

Step 4: Maximize the Area Expression

To maximize the area \( A \), we need to find the optimal values of \( \alpha \) and \( \beta \). Given \( \gamma = 2\pi - (\alpha + \beta) \), the function simplifies when \( \alpha = \beta = \gamma = \frac{2\pi}{3} \). This configuration is an equilateral triangle.

Step 5: Verify the Solution

If \( \alpha = \beta = \gamma = \frac{2\pi}{3} \), we have:\[ \sin \alpha = \sin \beta = \sin \gamma = \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} \]Substituting back, the area is:\[ A = \frac{1}{2} r^{2} [\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} - (-\frac{\sqrt{3}}{2})] = \frac{1}{2} r^{2} \times \frac{3\sqrt{3}}{2} = \frac{3\sqrt{3}}{4} r^{2} \]This confirms that the maximum area corresponds to an equilateral triangle.

Key Concepts

Calculus Problem Solving

Calculus is a powerful tool used to solve various mathematical problems, especially those involving optimization, such as finding maximum or minimum values. When tackling a problem, the first step involves understanding what is being asked.
In the scenario of maximizing the area of a triangle inscribed in a circle, calculus can be applied to find the shape and size of the triangle that results in the largest area. This involves exploring parameters that describe the shape, such as central angles in this case, and then using derivatives to find where the maximum area occurs.
Derivatives help identify points where the area changes from increasing to decreasing, revealing local maximums. When solving these types of problems, we ensure our found maximum is not just local, but the global maximum by analyzing the problem constraints.
With the right approach, calculus provides a systematic method to not only solve, but also verify the accuracy of solutions in geometric contexts.

Maximizing Area

Maximizing the area of a triangle inscribed in a circle is a classic problem that showcases the power of geometric intuition and mathematical analysis. The formula for the area of the triangle, in terms of the central angles, is:

• \( A = \frac{1}{2} r^{2} [\sin \alpha + \sin \beta - \sin(\alpha + \beta)] \)

To maximize this area, we focus on the angles \( \alpha \) and \( \beta \) and express them in terms of each other through the relationship \( \alpha + \beta + \gamma = 2\pi \).
This simplification reduces the formulas and allows easier manipulation to find the optimal configuration. Here, the calculus is used to take derivatives with respect to the angles and set them equal to zero to find the critical points responsible for maximizing the area.
Through this process, we find that when the triangle is equilateral, meaning all angles are equal, the area is maximized. This result beautifully ties together the concepts of symmetry in geometry and optimization through calculus.

Central Angles of a Triangle

Central angles are crucial in understanding triangles inscribed in circles. They are the angles whose vertex is at the circle's center. For an inscribed triangle, these angles govern the lengths of the sides of the triangle.

• Each central angle subtends a side of the triangle.
• The sum of the central angles \( \alpha + \beta + \gamma = 2\pi \).

These relationships are fundamental to ensure the triangle fits perfectly within the circle. By adjusting these angles, we can influence the triangle's shape and area.
Mathematically, using the sine of these angles connects to the formula for the area of the triangle. This importance is highlighted in solving for the maximum area configuration, where strategic analysis of angle changes reveals the equilateral triangle as a specially optimal case.
Central angles reflect the symmetry and provide a different perspective in geometric problem-solving, demonstrating how properties of circles and triangles intertwine.

Equilateral Triangle

The equilateral triangle emerges as the hero of the problem, offering the maximum area when inscribed within a circle. By definition, an equilateral triangle has all three sides and all three angles equal. In context:

• All central angles \( \alpha = \beta = \gamma = \frac{2\pi}{3} \).
• Sides of equal length result in maximum symmetry and balance.

What is remarkable about equilateral triangles inscribed in circles is that this symmetry leads to the optimal use of given space, ensuring the largest possible area is contained.
In terms of calculus and geometry, it showcases intuition meeting exact science. By confirming this through calculations, using trigonometric values such as the sine function, we see the formula resolve into an elegant solution uniquely pointing to the equilateral shape.
Thus, an equilateral triangle not only symbolizes balance in geometry but also reflects the optimal arrangement for this specific problem of maximizing area inscribed in a circle.