4. (Average) A vertical pole has a cross section of a regular hexagon with a side length of 1 foot. If a dog on a leash is tied onto one vertex, and the leash is three feet long, determine the total area that the dog can walk on.
Answer: $\frac{23}{3} \pi$ square feet
Solution: The total area is indicated by the lightly shaded region below.
$$
\begin{array}{l}
\frac{240}{360} \pi(3)^{2}+2\left(\frac{60}{360}\right) \pi(2)^{2}+2\left(\frac{60}{360}\right) \pi(1)^{2} \\
\frac{2}{3}(9 \pi)+\frac{1}{3}(4 \pi)+\frac{1}{3}(\pi)=\frac{23}{3} \pi
\end{array}
$$