Problem 53 - Algebra and Trigonometry

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

The first ranger is approximately \(\frac{7}{\cos(28^\circ)}\) miles away from the fire. Make sure to round to the nearest tenth of a mile as the problem instructs.

Step 1: Setup

From the given problem, one can form a right triangle. The first ranger is directly to the north of the fire, creating the vertical leg of a right triangle. The second ranger is to the east of the first ranger by 7 miles which forms the horizontal leg. The line from the second ranger to the fire, with a bearing of S 28° W, forms the hypotenuse.

Step 2: Use of Trigonometric Ratios

From the second ranger's perspective, the angle between the horizontal leg and the hypotenuse is 28 degrees. This now forms a right triangle where we know that the horizontal side is 7 miles (adjacent to the angle), and we need to find the length of the hypotenuse. We can use the cosine function to express the relation between the hypotenuse and the adjacent side. Let \(x\) represent the length of the hypotenuse (distance from the fire to the second ranger). Hence, we have the equation \(\cos (28^\circ) = \frac{7 \, \text{miles}}{x}\)

Step 3: Solve for Unknown Distance

By cross multiplication, we get \(x = \frac{7 \, \text{miles}}{\cos(28^\circ)}\). Solve this expression to find the value of \(x\).