Problem 51 - Precalculus Essentials

Question

A boat leaves the entrance to a harbor and travels 150 miles on a bearing of \(\mathrm{N} 53^{\circ} \mathrm{E}\). How many miles north and how many miles east from the harbor has the boat traveled?

Step-by-Step Solution

Answer

The boat has traveled approximately 113.13 miles East and 120.21 miles North from the harbor.

Step 1: Identify the Given Information

It is given that a boat travels 150 miles on a bearing of \(N 53^{\circ} E\). The bearing indicates the direction the boat is travelling, North East in this case, and the angle \(53^{\circ}\) is the angle made with North towards East. We need to calculate how many miles the boat has traveled North (upwards) and how many miles it has traveled East (rightwards).

Step 2: Calculating the Eastward Distance

We can represent our problem as a right-angled triangle, where the hypotenuse is the total distance of the boat, i.e., 150 miles, from the harbor. The eastward travel distance (along the base of triangle) can be found by using Cosine rule: Cosine(angle) = Adjacent / Hypotenuse. Here, the 'Adjacent' side represents distance East and the Hypotenuse is total distance i.e. 150 miles. So, \[Eastward Distance = 150 \times Cos(53^{\circ})\]

Step 3: Calculating the Northward Distance

Now, let's calculate the Northward distance (height of the triangle). We'll use Sine rule for this: Sine(angle) = Opposite / Hypotenuse. Here, the 'Opposite' side represents distance North and the Hypotenuse is again total distance (i.e., 150 miles). So, \[Northward Distance = 150 \times Sin(53^{\circ})\]

Step 4: Evaluation

By evaluating the expressions found in steps 2 and 3, the eastward (rightwards) and northward (upwards) distances can be found. Don't forget to round your answers to suitable decimal places if necessary.