Because of the wind over the meadow, the moth will require more energy to fly straight to C, than to fly straight to an intermediate point B and then along the woodline to the point C.

Suppose the moth requires 1 unit of energy per meter when flying along the woodline and e units of energy per meter (e ³ 1) when flying over the open meadow. Your goal is to determine the coordinates of the point B that minimize the amount of energy required by the moth.

1. Let d represent the distance from O to C. Find that distance.
2. Let x represent the distance from O to the variable point B. Find a function E(x) that gives the total energy required by the moth to fly from A to B to C.
3. Determine E¢(x) (assuming e is constant). Find those x-values for which E¢(x) = 0.
4. Suppose e = 2.
   1. Sketch the graph of E for 0 £ x £ d.
   2. Find the corresponding x-value that minimizes E.
   3. At what angle must the moth fly?
5. If e = 1, the moth will fly straight to C. Explain why this is true. What if e = 1.2?
6. Find the greatest value of e for which the moth will fly straight to C.