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A farmer wants to create an enclosure for his chickens, so he has purchased 28 meters of chicken coop wire mesh.
1. Initially the farmer considers making a rectangular enclosure.
1. Complete the following table to show all the possible rectangular enclosures with sides of at least 4 m he can make with the 28 m of mesh. The sides of the enclosure are always a whole number of metres.
2. What is the name of the shape that gives the maximum area?
The farmer wonders what the area will be if instead of a rectangular enclosure he uses an equilateral triangular enclosure.
2. Show that the area of the triangular enclosure will be $\frac{196 \sqrt{3}}{9} $.
Next, the farmer considers what the area will be if the enclosure has the form of a regular pentagon.
The following diagram shows a reqular pentagon.
Let O be the centre of the regular pentagon. The pentagon is divided into five congruent isosceles triangles and angle $A \widehat{O}$ B is equal to $\theta $ radians.
3. 1. Express $\theta$ in terms of $\pi$ .
2. Show that the length of OA is $\frac{14}{5}$$ \operatorname{cosec}\left(\frac{\pi}{5}\right) \mathrm{m}$ .
3. Show that the area of the regular pentagon is $\frac{196}{5} \cot \left(\frac{\pi}{5}\right) \mathrm{m}^{2} $.
Now, the farmer considers the case of a regular hexagon.
4. Using the method in part (c), show that the area of the regular hexagon is
$\frac{196}{6} \cot \left(\frac{\pi}{6}\right) \mathrm{m}^{2}$
The farmer notices that the hexagonal enclosure has a larger area than the pentagonal enclosure. He considers now the general case of an n -sided regular polygon. Let A_{n} be the area of the n -sided regular polygon with perimeter of 28 m .
5. Show that $A_{n}=\frac{196}{n} \cot \left(\frac{\pi}{n}\right)$ .
6. Hence, find the area of an enclosure that is a regular 14-sided polygon with a perimeter of 28 m . Give your answer correct to one decimal place.
7. 1. Evaluate $\lim _{n \rightarrow \infty} A_{n}$
2. Interpret the meaning of the result of part (g) (i).
