Problem 53 - Precalculus Essentials

Question

A forest ranger sights a fire directly to the south. A second ranger, 7 miles east of the first ranger, also sights the fire. The bearing from the second ranger to the fire is \(\mathrm{S} 28^{\circ} \mathrm{W}\). How far, to the nearest tenth of a mile, is the first ranger from the fire?

Step-by-Step Solution

Answer

After evaluating the expression, the first ranger's approximate distance from the fire is found to be 8.0 miles. Further specific value to the nearest tenth of a mile would be dependent on the exact value of \(\sin(62)\).

Step 1: Understand the Problem

The problem involves two rangers spotting fire from their different positions in a forest. The coordinates are related to cardinal directions: North, East, South and West. As the bearing from the second ranger is given as S28°W, this means that the angle formed between the south line (line of sight of ranger 1) and the line of sight of Ranger 2 is 28 degrees west of south.

Step 2: Draw a Diagram

Start by drawing a diagram. Place the first ranger at the origin of a coordinate plane. Mark the second ranger's position 7 miles to the east on the x-axis. Draw a line due south from the first ranger to represent their view of the fire. From the second ranger’s location, draw a line, towards southwest (28 degrees west of south). The place where this line cuts the vertical line denotes the fire's location.\nThis forms a non-right triangle with the 7-mile distance between the rangers as one side, the unknown distance from ranger 1 to the fire as another side, and 90° – 28° = 62° angle.

Step 3: Apply Law of Sines

We have two angles (62° and 90°), and one side of the triangle (7 miles) known. For finding the third side (distance between the fire and ranger 1, let's say 'd'), we can use the Law of Sines which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle. Writing the equation for our triangle: \(\frac{d}{\sin(90)} = \frac{7}{\sin(62)}\).\n

Step 4: Solve for 'd'

Solving the equation for 'd' gives us: \(d = \frac{7 \cdot \sin(90)}{\sin(62)}\)

Step 5: Evaluate Answer

Finally, evaluate the value of 'd' to the nearest tenth. The sine of 90 degrees is 1. It's a common trigonometric value, so the final distance d is approximately \(d = \frac{7}{\sin(62)}\). Calculate the numerical value and round off to the nearest tenth of a mile.