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Beehives In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle \(\theta\) is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area \(S\) is given by \(S=6 s h-\frac{3}{2} s^{2} \cot \theta+\left(3 s^{2} \sqrt{3} / 2\right) \csc \theta\) where \(s,\) the length of the sides of the hexagon, and \(h,\) the height, are constants. (a) Calculate \(d S / d \theta\) (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of \(s\) and \(h\) .

Step by step solution

01

Differentiate the Surface Area Formula

To find \( \frac{dS}{d\theta} \), differentiate the given surface area formula with respect to \( \theta \). The surface area formula is:\[S = 6sh - \frac{3}{2}s^2 \cot \theta + \left(\frac{3s^2 \sqrt{3}}{2}\right) \csc \theta\]Differentiate each term:1. The first term \( 6sh \) is a constant with respect to \( \theta \), so its derivative is 0.2. The derivative of \( -\frac{3}{2} s^2 \cot \theta \) is \( \frac{3}{2} s^2 \csc^2 \theta \) using the derivative \( \frac{d}{d\theta}(\cot \theta) = -\csc^2 \theta \).3. The derivative of \( \left(\frac{3s^2\sqrt{3}}{2}\right) \csc \theta \) is \(-\left(\frac{3s^2\sqrt{3}}{2}\right) \csc \theta \cot \theta \) using \( \frac{d}{d\theta}(\csc \theta) = -\csc \theta \cot \theta \).Combine these derivatives to get:\[\frac{dS}{d\theta} = \frac{3}{2}s^2 \csc^2 \theta - \left(\frac{3s^2\sqrt{3}}{2} \right) \csc \theta \cot \theta\]

02

Find the Preferred Angle

To determine what angle bees should prefer, set \( \frac{dS}{d\theta} = 0 \) and solve for \(\theta\). \[\frac{3}{2} s^2 \csc^2 \theta - \frac{3s^2\sqrt{3}}{2} \csc \theta \cot \theta = 0 \]Factor out common terms:\[\frac{3}{2}s^2 \csc \theta ( \csc \theta - \sqrt{3} \cot \theta ) = 0 \]Therefore, for the expression to equal zero, \[\csc \theta - \sqrt{3} \cot \theta = 0 \]This simplifies to:\[\csc \theta = \sqrt{3} \cot \theta \]Knowing \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), substitute these in:\[\frac{1}{\sin \theta} = \sqrt{3} \frac{\cos \theta}{\sin \theta} \]\[1 = \sqrt{3} \cos \theta \]\[\cos \theta = \frac{1}{\sqrt{3}} \]\[\theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

03

Determine the Minimum Surface Area

Substitute \( \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \) back into the original surface area formula \( S \). Using the identity \( \csc \theta = \sqrt{1 + \cot^2 \theta} \):\[\cot \theta = \sqrt{2} \]\[\csc \theta = \sqrt{3} \]Substitute these values in:\[S = 6sh - \frac{3}{2}s^2 (\sqrt{2}) + \left(\frac{3s^2\sqrt{3}}{2}\right)(\sqrt{3})\]\[S = 6sh - \frac{3\sqrt{2}}{2}s^2 + \frac{9}{2}s^2\]This yields the simplified expression for minimal \( S \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hexagonal Prism

A hexagonal prism is a three-dimensional solid shape that consists of two parallel hexagons and six rectangular faces connecting the sides of these hexagons. In the context of a beehive, each cell is structured as a hexagonal prism with an open end and a trihedral angle at the other.
Its geometry is fascinating because it combines elements of both three-dimensional and hexagonal properties. This structure is optimal for the storage of honey and brood because it allows efficient use of space and wax. The regular hexagonal prism ensures each cell is identical, maximizing the utility of the materials used.
The prisms are consistent in shape with defined parameters such as side length and height, critical for further calculations in optimization problems.

Optimization

Optimization is the process of making something as efficient or effective as possible. In terms of the beehive problem, optimization involves minimizing the surface area of each hexagonal prism cell for a given side length and height. This is essential as it economizes on wax usage in construction.
By finding the optimal apex angle, bees minimize the amount of wax required to build each cell, allowing them to focus resources on other tasks. This reveals nature's intrinsic capability to solve complex optimization problems. The method used by bees shows an elegant balance between the structure's complexity and the economy of resources, a considerable evolutionary advantage.

Surface Area Formula

The surface area formula for the hexagonal prism cell is given as: \[ S = 6sh - \frac{3}{2}s^2 \cot \theta + \frac{3s^2\sqrt{3}}{2} \csc \theta \]This formula calculates the total external surface enclosing the cell.
Here:

• \(s\) represents the side length of the hexagon's base.
• \(h\) is the height of the prism.
• \(\theta\) is the angle at the trihedral apex.

This formula incorporates trigonometric functions such as cotangent (\(\cot\)) and cosecant (\(\csc\)) to handle the angular component, which adds complexity due to the cell's geometric nature. Understanding the formula is key to solving for the optimal conditions in the context of bees and beehives.

Differentiation

Differentiation is a fundamental concept in calculus, used to find the rate at which a function is changing at any given point. For optimization, it reveals how the surface area of the beehive cell changes with respect to the apex angle \(\theta\).
By differentiating the surface area formula \( S \) with respect to \(\theta\), we determine how incremental changes in \(\theta\) affect the surface area \(S\). The derivative \( \frac{dS}{d\theta} \) results in:\[ \frac{dS}{d\theta} = \frac{3}{2}s^2 \csc^2 \theta - \frac{3s^2\sqrt{3}}{2} \csc \theta \cot \theta \]Setting this derivative to zero helps find the critical points where the surface area might be minimized, further optimizing the cell design.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They play a critical role when analyzing the geometry of the hexagonal prism cells in bee hives.
In the context of the surface area optimization problem, identities such as \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) simplify the derivatives and allow us to solve for \(\theta\).
For instance, solving \(\csc \theta = \sqrt{3} \cot \theta\) by substituting the trigonometric identities enables us to find the specific angle \(\theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\), optimizing the hive's construction to minimize wax usage.