A hexagonal prism has a height of 165 cm. Its two hexagonal faces are regular hexagons with sides of length 30 cm. Its other six faces are rectangles. A fly and an ant start at point X on the bottom face and travel to point Y on the top face. The fly flies directly along the shortest route through the prism. The ant crawls around the outside of the prism along a path of constant slope so that it winds around the prism exactly n + 1/2 times, for some positive integer n. The distance crawled by the ant is more than 20 times the distance flown by the fly.
What is the smallest possible value of n?

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 is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by

where s, the length of the sides of the hexagon, and h, the height, are constants.

(a) Calculate .

(b) What angle should the bees prefer?

(c) Determine the minimum surface area of the cell (in terms of s and h).

London misplaced his 14mm wrench, so he is 3D printing a replacement, just a card with a regular hexagon-shaped hole (regular means that the sides are all the same length). The 14mm of a hex wrench is the distance from one flat side to the flat side opposite; however, London’s 3D printing software requires the distance from one vertex (corner) to the opposite vertex. Draw a diagram of this scenario. Label the 14mm and the distance London needs to calculate.
Find the distance that London needs for his design. Round your answer to the nearest tenth of a mm. SHOW YOUR WORK.
Hint: If you connect each vertex with its opposite, then you have divided the hexagon into six triangles. What has to be true about the angles of those triangles?

7. A straight line passing through the point (8, 27) intersects the positive z-axis at the point P and the positive y-axis at the point Q. What is the shortest possible distance between P and Q.