Problem 46 - Algebra and Trigonometry

Question

You are on a fishing boat that leaves its pier and heads east. After traveling for 30 miles, there is a report warning of rough seas directly south. The captain turns the boat and follows a bearing of \(\mathrm{S} 45^{\circ} \mathrm{W}\) for 12 miles. a. At this time, how far are you from the boat's pier? Round to the nearest tenth of a mile. b. What bearing could the boat have originally taken to arrive at this spot?

Step-by-Step Solution

Answer

a. The boat is approximately \(\sqrt{(30 - 12*\cos(45))^2 + (12*\sin(45))^2}\) miles from the pier. b. He could have taken the bearing of either \(90 - θ\) or \(450 - θ\), depending upon the value of θ.

Step 1: Breaking down the journey

Break down the boat's journey into two legs. First, the boat travels 30 miles East. Next, the boat turns and travels on a bearing of S 45° W for 12 miles. Make a sketch to visualize this.

Step 2: Resolve the second leg of journey into components

The second leg of the journey is not directly south or west, it is in the middle. So split it up into a westward segment and a southward segment. This can be done using trigonometry, a westward distance of \(12*\cos(45)\) miles and a southward distance of \(12*\sin(45)\) miles.

Step 3: Calculate total distance travelled in each direction

Since the east-west distances are in opposite directions, subtract the westward distance from the eastward distance, getting \(30 - 12*\cos(45)\) miles. The total southward distance is just \(12*\sin(45)\) miles as there was no initial southward travel.

Step 4: Calculate total distance using the Pythagorean Theorem

Distance from the pier is found out by using Pythagorean theorem: \(\sqrt{(30 - 12*\cos(45))^2 + (12*\sin(45))^2}\).

Step 5: Determine overall direction of travel

To find out the overall direction, which is the angle θ, we can use the formula for tangent, \(tan(θ)=\frac{opposite side}{adjacent side}=\frac{12*\sin(45)}{30 - 12*\cos(45)}\). Apply inverse tangent function to find θ.

Step 6: Convert the angle into bearing

Remembering that standard compass bearings start from North and move clockwise, the bearing is \(90 - θ\) if θ is less than 90, if it is greater, the bearing is \(450 - θ\).