Problem 24 - College Trigonometry

Question

Along a long, straight stretch of mountain road with a \(7 \%\) grade, you see a tall tree standing perfectly plumb alongside the road. \({ }^{14}\) From a point 500 feet downhill from the tree, the angle of inclination from the road to the top of the tree is \(6^{\circ}\). Use the Law of Sines to find the height of the tree. (Hint: First show that the tree makes a \(94^{\circ}\) angle with the road.)

Step-by-Step Solution

Answer

The height of the tree is approximately 53.15 feet.

Step 1: Understanding the Slope and Tree

The road has a 7% grade, meaning for every 100 feet horizontal, the road ascends 7 feet vertically. Since the tree is plumb (vertical), it forms a right triangle with the road and a line perpendicular from the base of the tree to the road. The angle between the tree and the sloped road should be calculated.

Step 2: Determine Angle with the Tree

The angle of the road grade is found using the inverse tangent of the grade. \[\theta = \tan^{-1}(0.07) \approx 4^{\circ}\]The angle between the tree (vertical) and the sloped road is then:\[90^{\circ} + 4^{\circ} = 94^{\circ}\]

Step 3: Create a Triangle and Apply Law of Sines

We have a triangle with angles of \( 94^{\circ} \), \( 6^{\circ} \) and the third angle (angle with the tree) is \(180^{\circ} - (94^{\circ} + 6^{\circ}) = 80^{\circ}\). The side opposite the 80° angle (500 feet) is known. Use the Law of Sines:\[\frac{h}{\sin(6^{\circ})} = \frac{500}{\sin(80^{\circ})}\]

Step 4: Solve for the Height of the Tree

Rearrange to solve for \(h\):\[h = \frac{500 \times \sin(6^{\circ})}{\sin(80^{\circ})}\]Calculate the height:\[h \approx \frac{500 \times 0.1045}{0.9848} \approx 53.15 \text{ feet}\]

Key Concepts

Trigonometric Applications

Trigonometry is a powerful branch of mathematics that deals with the relationships between angles and sides of triangles. Its applications are vast, ranging from engineering and physics to everyday problem-solving scenarios involving angles and distances. In this context, we are using trigonometry to find the height of a tree along a sloped mountain road by applying the Law of Sines. This problem showcases the practical use of trigonometry to calculate unknown distances and angles when direct measurement is not possible. The key here is forming a triangle using the given angles, known side lengths, and the incline of the road:

• Angle of inclination of the roadway, θ, calculated using inverse tangent due to the road's slope.
• Forming a triangle with the observed angle of inclination and solving for unknown height using the Law of Sines.

The Law of Sines provides a way to connect these angles with the known and unknown sides, letting us solve for the height without needing direct vertical measurement. This is a perfect example of how trigonometric principles are practically used.

Angle of Elevation

The angle of elevation is a key concept here, which is the angle formed between the horizontal plane and the line of sight to an object above the horizontal plane. In our problem, it is the angle formed from a point on the road, 500 feet downhill, to the top of the tree at 6°. This angle tells us how much the line of sight tilts upwards when viewing the top of a object from a lower point. Understanding how to calculate angles of elevation is essential when applying concepts like the Law of Sines or solving other trigonometric problems:

• Measure from the horizontal upwards to an object.
• Use angles of elevation in determining comparative heights in problems involving elevation.

Knowing this angle and the slope of the road allows us to set up our triangle properly, creating a relationship between the observed angle and the rest of the problem's geometry. It was a crucial step that led to our ability to apply the Law of Sines to find the height of the tree.

Solving Right Triangles

In this exercise, solving right triangles involves understanding the angles in relation to the given side lengths and using trigonometric relationships to find unknown values. The tree forms a right angle to the base, providing a starting basis for our calculations. By finding the angle between the slope and the tree ( 94°), we've essentially divided the space into two right triangles:

• The triangle involving the tree's height, the distance from the tree along the road, and the sloped road itself.
• An auxiliary triangle helping calculate the incline angle given the road's slope percentage.

The problem also includes non-right angles formed at the base of the tree. Using these, the Law of Sines helps resolve the issues with angles and sides of different measurements. For example, we calculated the needed angles and applied sin functions to isolate and find the height of the tree. Understanding these steps clarifies how right triangles can be used effectively in non-standard configurations to solve real-world problems.