Problem 69 - 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 Was 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 value of \(\theta\) that minimizes \(W\) is approximately \(\frac{2\pi}{3}\) radians.

Step 1: Understand the Function

We are given the function for the amount of wax in the cell, which is \(W = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\). The goal is to find the apex angle \(\theta\) that minimizes \(W\). This function is defined over the interval \(0 < \theta < \pi\).

Step 2: Graph the Function

Using a graphing device or software (such as Desmos or GeoGebra), plot the function \(W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\) over the interval \(0 < \theta < \pi\). The graph will help visualize how \(W\) changes with respect to \(\theta\) and will highlight the behavior of the function near the minimum.

Step 3: Analyze the Graph

Examine the graph you plotted to identify the point where the function \(W\) reaches its minimum value. Look for the lowest point on the curve of the function \(W\) within the domain \(0 < \theta < \pi\). This will help identify the approximate value of \(\theta\) that minimizes \(W\).

Step 4: Identify the Minimum Value

From the graph, determine the exact value or approximate numerical value of \(\theta\) at which \(W\) is minimized. Biologically, bees have evolved to select this angle, which is typically around \(120^\circ\) or \(\frac{2\pi}{3}\) radians.

Key Concepts

Graphing Functions

Understanding the behavior of a function graphically can be incredibly helpful. This is particularly true when dealing with trigonometric functions, such as those involving cotangent (\(\cot\)) and cosecant (\(\csc\)). To graph the function \(W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\), you'll need to consider its characteristics over the interval \(0 < \theta < \pi\).

A graphing device, such as an online tool like Desmos or GeoGebra, can be used to plot this function effectively. This visual representation will allow you to examine how the amount of wax \(W\) varies as the apex angle \(\theta\) changes.

By interpreting the graph, you can identify trends and find the point of intersection that represents the minimum value of the function. Remember, this process involves looking for the lowest point on the curve, which corresponds to the variable \(\theta\) that results in the minimum amount of wax used.

Apex Angle Optimization

The concept of apex angle optimization ties directly into the bees' natural abilities. Bees, through evolution, have found a meaningful balance for the apex angle \(\theta\) to use minimal wax. This angle is crucial because it influences the efficiency and resource usage within the hive.

Mathematically, the task is to find the value of \(\theta\) where \(W\) reaches its minimum. As we analyze the graph of the function \(W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\), we look for the angle where the curve dips to its lowest point. Engineers and mathematicians approach this with tools such as calculus for verification, but the graph serves as a first visualization to zone in on the optimal value.

It's fascinating how nature optimizes processes without deliberate calculation, arriving at angles like approximately \(120^\circ\) or \(\frac{2\pi}{3}\) radians, which is acknowledged as the optimal value by various engineers and scientists.

Wax Minimization

Wax minimization is directly related to bees' survival and efficiency. They instinctively construct their hexagonal cells in a way that uses the least amount of wax, which supports more substantial and sustainable hives. This involves choosing an apex angle \(\theta\), which reflects in the equation \(W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\), where \(W\) should be minimized.

Biologically, this behavior is essential because wax is resource-intensive and energetically costly to produce. By minimizing wax use, bees conserve energy and materials, which directly contributes to the hive's efficiency and resource management.

The mathematical model reflects this biological imperative and is a great example of how mathematical analysis and optimization principles are mirrored in the natural world. By understanding and minimizing \(W\), not only do students learn about optimization, but they also gain insights into the elegant efficiency of nature's designs.