Final answer:
To calculate the bearings from the cliff to the island, we apply trigonometry to the path taken by the puffin forming a right triangle. The bearing from the cliff to the island is found to be North 55° East, determined by the arctangent of the northward to eastward distance ratio.
Step-by-step explanation:
The stated problem is a trigonometric exercise dealing with bearings and vector addition. A puffin flies from a cliff to an island following a path of 38km east and then 54km due north. The bearings can be found by considering this path forms a right triangle with the distance east as one leg and the distance north as the other leg. To find the bearings from the cliff to the island, we use the concept of bearings which indicates direction with respect to the north.
To calculate the bearings, we consider the angle of bearing from the north, going clockwise. We can determine the angle using trigonometry, specifically the tangent function, which relates the opposite side (54km north) and adjacent side (38km east) of the right triangle. The angle θ can be found using the arctangent (tan⁻¹) of the ratio 54/38.
θ = tan⁻¹(54/38)
Calculating this angle provides us with the deviation from the north towards the east. Hence, the bearing is North θ East, where θ is the angle we calculated. Solving this angle we get approximately 55 degrees. The correct answer is therefore North 55° East.