Moana (an expert sailor) is sailing from her home to a nearby island which is 215 miles north and 33 miles west. There is a constant ocean current of 0.50 knots moving from west to east. Moana can sail her boat at a cruising speed of 5.5 miles per hour in still water.

Question

1. What angle should Moana sail to get to the island?

2. How long will it take her to get there?

3. Moana’s friend Maui is floating on a raft that is moving with the ocean
current (meaning, he is not traveling with respect to the water). Would
Maui see Moana’s boat traveling with a speed faster than, slower than, or
equal to what an observer standing on the shore would see? Explain your
reasoning.

Answer 1

Sail At angle T west of North
speed North = 5.5 cos T
speed west = 5.5 sin T- 0.5
now sail for t hours
speed north * t = 215
speed west *t = 33
so
(5.5 cos T) t = 215
(5.5 sin T- 0.5) t = 33
so
t = 39 / cos T
(5.5 sin T- 0.5) 39 / cos T = 33
5.5 sin T - 0.5 = (33/39) cos T = 0.846 cos T
5.5 sin T - .846 cos T - 0.5 = 0
5.5 sqrt (1-cos^2 T)= .846 cos T + 0.5
sqrt (1 - cos^2 T) = 0.154 cos T + .091
1 - cos^2 T =.0237 cos^2 T + .0474 cos T + .00828
0 = 1.0237 cos^2 T + .0474 cos T - 0.992
solve quadratic
https://www.mathsisfun.com/quadratic-equation-solver.html
cos T = .962
so
T = 15.85
so sail 15.85 deg west of north

Answer 2

now go back and get t
faster, headed toward