Problem 105 - Algebra and Trigonometry

Question

Sawing a Wooden Beam A rectangular beam is to be cut from a cylindrical log of diameter 20 in. (a) Show that the cross-sectional area of the beam is modeled by the function $$A(\theta)=200 \sin 2 \theta$$ where \(\theta\) is as shown in the figure. (b) Show that the maximum cross-sectional area of such a beam is \(200 \mathrm{in}^{2} .\) [Hint: Use the fact that sin \(u\) achieves its maximum value at \(u=\pi / 2.1\)

Step-by-Step Solution

Answer

Maximum area is 200 in² when \( \theta = \pi/4 \).

Step 1: Understanding the Problem

We have a cylindrical log with a diameter of 20 inches. We need to find the beam's cross-sectional area that can be cut from it, expressed in terms of the angle \( \theta \). This involves geometric analysis since the beam is inscribed within the circular cross-section.

Step 2: Expressing the Geometry

Visualize a circle with diameter 20 inches inside which a rectangle (beam) is inscribed. The rectangle's diagonal is equal to the circle's diameter, so the diagonal length is 20 inches. The angle \( \theta \) is formed between the beam's base and the circle's center.

Step 3: Using Trigonometry in the Circle

The diagonal of the rectangle also forms two right triangles within the circle. The trigonometric identity for a right triangle within the circle is \( \sin \theta = \frac{opposite}{hypotenuse} \). In this case, the 'opposite' part of the angle \( 2\theta \) corresponds to half the rectangle's width.

Step 4: Derive the Function for Area

The width of the beam can be expressed as \( 20 \sin \theta \) and the height as \( 20 \cos \theta \) (opposite and adjacent sides of the angle respectively). Therefore, the area of the beam is \( A(\theta) = 20 \sin \theta \times 20 \cos \theta = 400 \sin \theta \cos \theta \).

Step 5: Using Double Angle Formula

Apply the trigonometric identity \( \sin 2\theta = 2 \sin \theta \cos \theta \). Thus, the equation becomes \( A(\theta) = 400 \sin \theta \cos \theta = 200 \sin 2\theta \), which matches the given function for the area.

Step 6: Finding the Maximum Area

The sine function, \( \sin 2\theta \), reaches its maximum value of 1 when \( 2\theta = \frac{\pi}{2} \). Therefore, \( \theta = \frac{\pi}{4} \). At this point, \( A(\theta) = 200 \times 1 = 200 \text{ in}^2 \), confirming that the maximum cross-sectional area is indeed 200 square inches.

Key Concepts

Cross-sectional Area

The cross-sectional area is a measure of the two-dimensional space within the boundary of a section cut through an object. In this exercise, we focus on a rectangular beam cut from a cylindrical log. The beam's cross-sectional area is what we observe when we slice through the beam perpendicularly to its length.
The function provided, \(A(\theta) = 200 \sin 2\theta\), gives the area in terms of the angle \(\theta\). Here, \(\theta\) is the angle formed between the base of the beam and the circle's center. The area function is derived from the geometry of the inscribed rectangle within the circle.
This classical problem involves maximizing the area possible from the circular log. The function highlights how the area changes with \(\theta\), utilizing a trigonometric function to express the effect of \(\theta\) on the beam's dimensions.

Circular Geometry

Circular geometry plays a crucial role in understanding how objects fit inside circles. Imagine a circle having a diameter of 20 inches. Within this circle, a rectangle can be inscribed such that its diagonal coincides with the circle's diameter. This setup forms our cylindrical log beam example.
The diagonal acts as the hypotenuse of two identical right triangles formed inside the circle. The use of diameter as a diagonal ensures the entire beam remains within the circle's boundary, maximizing space usage within the circle. Understanding this aspect helps appreciate how geometric properties determine relationships between angles and line segments.
The idea here is to interpret angles, triangles, and other geometric elements to express properties such as area using circular constraints. The diameter and angles \(\theta\) and \(2\theta\) play pivotal roles in calculations built around the circle's inherent symmetry.

Trigonometric Identities

Trigonometric identities are powerful tools that allow for simplification and transformation of complex geometric and algebraic expressions.
In the provided exercise, the double angle identity \( \sin 2\theta = 2 \sin \theta \cos \theta \) is essential. It simplifies the expression for the cross-sectional area from \(400 \sin \theta \cos \theta\) to \(200 \sin 2\theta\). This transformation utilizes the inherent properties of trigonometric functions to derive simpler solutions.
Understanding identities like \( \sin(2\theta) \) extends beyond solving equations. It benefits from recognizing how angles and trigonometric functions interrelate across the circle's geometry. This base knowledge is vital for navigating calculations and predicting the behavior of trigonometric functions in varied contexts.