Problem 67 - Algebra and Trigonometry

Question

Beehives In a beehive each cell is a regular hexagonal prism, as shown in the figure. The amount of wax \(W\) in the cell depends on the apex angle \(\theta\) and is given by $$ W=3.02-0.38 \cot \theta+0.65 \csc \theta$$ Bees instinctively choose \(\theta\) so as to use the least amount of wax possible. (a) Use a graphing device to graph \(W\) as a function of \(\theta\) for \(0<\theta<\pi\) (b) For what value of \(\theta\) does \(W\) have its minimum value? [Note: Biologists have discovered that bees rarely deviate from this value by more than a degree or two.]

Step-by-Step Solution

Answer

The minimum value of \( W \) occurs at \( \theta \approx 2.6 \) radians.

Step 1: Set Up Function for Graphing

The function to be graphed is given as \( W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta \). We will need to graph this function for the domain \( 0 < \theta < \pi \). Ensure that your graphing tool is set to radian mode, as \( \theta \) is typically in radians.

Step 2: Describe Key Features of the Function

The function \( W(\theta) \) involves trigonometric terms \( \cot \theta \) and \( \csc \theta \), which have vertical asymptotes where they are undefined, specifically at \( \theta = 0 \) and \( \theta = \pi \). The graph will show large values approaching these points due to the behavior of the \( \cot \) and \( \csc \) functions.

Step 3: Graphing with Technology

Using a graphing calculator or software, input the function \( W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta \). Set the domain from just above \( 0 \) to just below \( \pi \). The plot should reveal how \( W \) varies with \( \theta \), highlighting any peaks and troughs in the curve.

Step 4: Identify the Minimum Value of W

Observe the graph to determine the minimum point of \( W(\theta) \). This is where the amount of wax required is least. Using zoom or features to calculate minima in graphing tools can help provide the most accurate \( \theta \).

Step 5: Confirm the Minimum Value Numerically

Use graphing tool functions to calculate the exact minimum numerically for precision. Graphing tools often have features to easily find and confirm the minimum value of a function in a given interval.

Key Concepts

Understanding Cotangent

The cotangent function, often denoted as \( \cot \theta \), is an important trigonometric concept, especially in optimization problems like the beehive exercise. It is defined as the reciprocal of the tangent function: \( \cot \theta = \frac{1}{\tan \theta} \). This means it is equal to \( \frac{\cos \theta}{\sin \theta} \). Cotangent has some unique properties:

• The function is periodic, with a period of \( \pi \).
• It is undefined when \( \sin \theta = 0 \), which occurs at integer multiples of \( \pi \) (excluding offsets where it is defined within the open interval \( 0 < \theta < \pi \)).
• The graph of \( \cot \theta \) exhibits vertical asymptotes at these points, where the function shoots off to infinity.

Understanding these behaviors helps in predicting how cotangent affects other functions, such as our wax function \( W(\theta) \). By knowing where the cotangent is undefined, we can infer regions of rapid change or large values for \( W(\theta) \). This makes it crucial in determining peril zones for wax usage.

Delving into Cosecant

The cosecant function, denoted as \( \csc \theta \), is another pillar of trigonometry, necessary for optimization tasks like reducing wax in a honeycomb cell. Defined as the reciprocal of the sine function, \( \csc \theta = \frac{1}{\sin \theta} \), it tends towards infinity wherever sine is zero. Cosecant has various properties:

• It is also periodic, with a period of \( 2\pi \).
• Similar to cotangent, it is undefined when \( \sin \theta = 0 \) due to its reciprocal nature.
• The graph features vertical asymptotes at multiples of \( \pi \), where the function value becomes extremely large.

In the wax function \( W(\theta) \), \( \csc \theta \) contributes to significant changes as it approaches these asymptotes. This insight into its graphical behavior is fundamental when minimizing functions analytically or graphically.

Graphing Trigonometric Functions

Graphing trigonometric functions often reveals intuitive insights into their behavior and their applications in real-world scenarios, like optimizing material usage in beehives. In our problem, the graph of \( W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta \) gives us a visual understanding of how bees might select \( \theta \) to minimize wax. While graphing:

• Set the calculator or software to the radian mode because \( \theta \) is typically measured in radians.
• Note that graphs of \( \cot \) and \( \csc \) exhibit asymptotes; hence the graph could show spikes as these functions become undefined.
• Adjusting the view to see the full period of \( \pi \) might help observe complete cycles of change.

Trigonometric graphs can initially be overwhelming, but identifying key features like maxima and minima can focus our analysis. By finding the minimum point on \( W(\theta) \)'s graph, we align our understanding with the bees' instinctive optimization strategy.

Optimizing with Trigonometry

Optimization in mathematics often seeks to find the point at which a particular function reaches its maximum or minimum value. In contexts like beehive construction, it involves determining how bees instinctively choose the angle \( \theta \) to minimize wax use. This involves working with the function \( W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta \). When optimizing a function using trigonometry:

• Start by observing the function's graph to locate the interval of interest. Here, \( 0 < \theta < \pi \).
• Identify the critical points where the function's derivative equals zero, indicating potential local minima or maxima.
• Ensure the function is continuous over the interval, acknowledging any points of discontinuity or asymptotes.

In this exercise, using numerical solutions from graphing tools can accurately pinpoint the optimal \( \theta \). This knowledge underscores the effectiveness of trigonometry in practical, natural settings.