Problem 69 - Precalculus Mathematics for Calculus

Question

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 wax is used when \( \theta \approx 120^\circ \) or \( \frac{2\pi}{3} \) radians.

Step 1: Understand the formula

The wax weight formula given is \( W = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta \). Our task is to examine how the weight \( W \) changes with respect to \( \theta \), where \( \theta \) is the apex angle.

Step 2: Understand the Interval

The interval for \( \theta \) is from just above 0 to just below \( \pi \) radians \((0 < \theta < \pi)\). This means bee angles typically used are somewhere between these values.

Step 3: Set up the graph

To graph this function \( W(\theta) \), use a graphing tool capable of handling trigonometric functions. Input \( W = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta \) and adjust the window to \( 0 < \theta < \pi \).

Step 4: Identify important features

While graphing, look for critical points where the function could have minimum or maximum values. Critical points are usually where the derivative is equal to zero or undefined.

Step 5: Analyze critical points

To find the minimum, calculate the derivative of \( W \) with respect to \( \theta \), \( W'(\theta) \). Set \( W'(\theta) = 0 \) or examine where it's undefined and solve for \( \theta \).

Step 6: Confirm the minimum value

Check the behavior of \( W \) at critical points using either the second derivative test or checking the values around these points. This confirms which corresponding \( \theta \) leads to a minimum wax usage.

Step 7: Note the biological observation

Biologists observe bees typically choose an angle slightly deviated by a degree or two from the ideal angle found by this graphing analysis.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to its sides. They are fundamental in studying periodic phenomena, such as waves, and they play a big role in optimization problems in areas including architecture, engineering, and indeed in the natural design of beehive cells.

The key trigonometric functions include sine (\(\sin\theta\)), cosine (\(\cos\theta\)), tangent (\(\tan\theta\)), cotangent (\(\cot\theta\)), secant (\(\sec\theta\)), and cosecant (\(\csc\theta\)). In the context of our hexagonal prism equation:

• \(\cot\theta\) is the cotangent function, defined as \(\cot\theta=\frac{1}{\tan\theta}\) or \(\frac{\cos\theta}{\sin\theta}\).
• \(\csc\theta\) is the cosecant function, defined as \(\csc\theta=\frac{1}{\sin\theta}\).

These functions transform an angle into a ratio of sides, which is key in solving our beehive cell problem, where we need to optimize the wax usage.

Critical Points

Critical points in a function are those points where the derivative is zero or undefined. Finding these points helps in identifying maxima, minima, or points of inflection of the function. In the case of our wax equation, identifying such points will tell us the values of \(\theta\) where the weight of the wax, \(W\), is minimized.

To locate critical points:

• Calculate the derivative of the function with respect to \(\theta\).
• Set the derivative equal to zero or examine where it may be undefined.

By solving these conditions, we identify candidate points for where the wax usage is either minimized or maximized, although in this scenario, we are particularly interested in minimizing wax use. It's a fascinating instance of using calculus to mimic nature's apparent instinctive efficiency!

Graphing Trigonometric Functions

Graphing is a powerful visual tool that allows us to better understand the behavior of trigonometric functions over a specified interval. For our beehive equation, graphing \(W(\theta) = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\) over the interval \(0<\theta<\pi\) helps visualize where wax usage is minimized.

The steps to graph this include:

• Using a graphing calculator or software, input the function equation.
• Set the interval from just above \(0\) to just below \(\pi\), capturing the range typical for bee behavior.
• Analyze the graph to visually identify dips, which indicate minimum points that correspond to potential wax savings for the bees.

Interpreting these points on the graph gives insight into both the mathematical and biological optimization happening within a beehive.

Derivatives in Calculus

Derivatives are essential in calculus for understanding how functions change. By examining the derivative of a function, you can determine critical points, which are vital for optimization tasks like our wax usage problem.

The derivative of the wax function, \(W'(\theta)\), gives us the rate of change of wax weight concerning the angle \(\theta\). To solve for minimizing wax, follow these steps:

• Compute the derivative \(W'(\theta)\) using rules for differentiating trigonometric functions such as \(\cot\theta\) and \(\csc\theta\).
• Set \(W'(\theta) = 0\) and solve for \(\theta\) to identify potential minima.
• Consider the second derivative or values around these \(\theta\) to confirm the minima via the second derivative test.

In this case, derivatives not only help determine the precise angle for optimal wax use but also underpin the biological genius observed in bees' natural design preferences.