The problems

1.

Prove that a parabola focuses incoming parallel rays upon a single point using your knowledge of calculus and trigonometry. Remember, when light reflects off a polished surface, the angle of incidence is equal to the angle of reflection (see figure above right).

Note: You might find this information in another book, and this is fine. However, you still must show all your work and explain every step.

2.

A roman galley is expected to pass 175 meters off a long straight beach. You have at your disposal as many soldiers and shields as you need, and the beach is 25 meters deep. To complicate matters, humidity from the Aegean Sea attenuates the focused energy as the light travels through damp air close to the surface of the water. If E is the focused energy in a vacuum, the amount of energy that reaches the focus in air is

$$\frac{E}{(p-150)^2}$$

where p>150 is the distance from the bottom of the parabola to the focal point. Find the optimal shape and placement of the parabola on the beach so that it focuses the most light energy on the galley.