Problem 56 - Algebra and Trigonometry

Question

A forest ranger is walking on a path inclined at \(5^{\circ}\) to the horizontal directly toward a 100 -foot-tall fire observation tower. The angle of elevation from the path to the top of the tower is \(40^{\circ} .\) How far is the ranger from the tower at this time?

Step-by-Step Solution

Answer

The ranger is approximately 119.66 feet from the tower.

Step 1 - Identify and Label the Given Information

Draw a right triangle where the fire observation tower is the vertical side (opposite side), the distance from the ranger to the base of the tower is the adjacent side, and the hypotenuse is the path the ranger is walking. Label the height of the tower as 100 feet, the angle of inclination of the path as \(5^{\circ}\), and the angle of elevation from the path to the top of the tower as \(40^{\circ}\).

Step 2 - Use Trigonometric Relationships

Using the triangle formed, let the distance from the ranger to the base of the tower be \(d\). Use the angle of elevation \(40^{\circ}\) to write the trigonometric relationship for the tangent function: \[ \tan(40^{\circ}) = \frac{100}{d} \].

Step 3 - Solve for the Distance

Isolate \(d\): \[ d = \frac{100}{\tan(40^{\circ})} \]. Calculate \( \tan(40^{\circ}) \) and solve for \(d\).

Step 4 - Calculate Using Calculator

\( \tan(40^{\circ}) \approx 0.839 \). Therefore, \[ d = \frac{100}{0.839} \approx 119.19 \text{ feet} \].

Step 5 - Adjust for Path Incline

Given that the path is inclined at \(5^{\circ}\), we need to adjust the distance \(d\) by the factor of the incline. Use the cosine of the incline angle: \[ \cos(5^{\circ}) \]. The adjusted distance \(d_{path}\) is \[ d_{path} = \frac{119.19}{\cos(5^{\circ})} \].

Step 6 - Final Calculation

\( \cos(5^{\circ}) \approx 0.996 \). Thus, \[ d_{path} = \frac{119.19}{0.996} \approx 119.66 \text{ feet} \].

Key Concepts

right triangle

A right triangle is a type of triangle that contains one angle measuring exactly 90 degrees. This means one of the three angles is always a right angle. The two sides forming this right angle are called the 'legs', and the side opposite the right angle is called the 'hypotenuse'. In the problem about the forest ranger and the tower, the right triangle is formed with the vertical tower as one leg and the horizontal distance from the ranger to the tower base as the other leg. The incline path the ranger walks on represents the hypotenuse. The Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the legs, is especially useful for solving right triangle problems.

angle of elevation

The angle of elevation is the angle formed by the line of sight when an observer looks upward to an object. In our problem, the angle of elevation is the angle between the path the ranger is walking on and the line of sight to the top of the tower. It is given as 40 degrees. This angle helps us use trigonometric functions, like tangent and cosine, to find the required distances. To visualize it better, imagine standing on flat ground and looking up at a tall building. The angle that your line of vision makes with the ground is the angle of elevation.

tangent function

The tangent function (tan) in trigonometry relates the opposite side to the adjacent side in a right triangle. For any angle θ, the tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side, given by: \[ \tan(\theta) = \frac{opposite}{adjacent} \] In our forest ranger problem, we used the angle of elevation (40 degrees) to set up the equation: \[ \tan(40^\text{°}) = \frac{100}{d} \] Here 100 feet is the opposite side, and 'd' is the adjacent side. Solving for 'd' gives us the horizontal distance from the ranger to the base of the tower.

cosine function

The cosine function (cos) describes the ratio of the adjacent side to the hypotenuse in a right triangle for any angle θ: \[ \theta = \frac{adjacent}{hypotenuse} \] In our problem, the path the ranger is walking on is inclined at an angle of 5 degrees. To find the true distance accounting for the incline, we adjust the calculated distance (d) through the cosine of 5 degrees. This adjustment is made to ensure we are considering the incline accurately, preventing overestimation or underestimation of distance. The relationship is set as: \[ d_{path} = \frac{d}{\text{cos}(5^\text{°})} \] After calculating, this correctly adjusts the ranger’s distance to about 119.66 feet.