Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

After a hurricane a tree is left standing but makes an angle of \(10^{\circ}\) with its former upright position. Suppose the tree is tilting away from the sun and casting a shadow 25 feet long. If the angle of elevation of the sun is \(30^{\circ}\), how long is the tree?

Step-by-Step Solution

Answer

The approximate length of the tree is 43.3 feet

Step 1: Establish the Situation

Consider the situation as a triangle. The shadow is the adjacent side, the tree is the opposite side, and the direct line from the sun to the end of the shadow is the hypotenuse. Hence, due to the angle of the sun, you're analyzing a special triangle (30-60-90 triangle). You know the length of the shadow (adjacent side) and need to find the length of the tree (opposite side).

Step 2: Apply Trigonometric Principles

To find the length of the tree, use the property of the 30-60-90 triangle, which states that the length of the side opposite to the 60-degree angle (length of the tree in this case) is \(\sqrt{3}\) times the length of the side opposite to the 30-degree angle (length of the shadow). Substituting the given values, you have the length of the tree as \(25 \times \sqrt{3}\).

Step 3: Calculate the Length of the Tree

After evaluating the expression from step 2, it's found that the length of the tree (rounded to the nearest tenths place) is approximately 43.3 feet. Note that the slight tilting of the tree by \(10^\circ\) is not significant enough to make any notable impact on the answer, so it's ignored in this basic trigonometric solution.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between a horizontal line and a line extending upwards to an object. In practical terms, it's when you look up from ground level to gauge the height of something above you. For this exercise, the angle of elevation of the sun is given as \(30^\circ\). This means if you were standing at the tip of the tree's shadow, this is the angle you'd look up at to see the sun.

• Understanding the angle of elevation helps in forming a right triangle.
• The horizontal distance is equal to the tree's shadow length.
• This angle is consistent with how sunlight hits objects on Earth.

Recognizing the angle of elevation is key as it allows you to apply trigonometric ratios to find the missing side lengths of the triangle.

Special Right Triangles

Special right triangles, like the 30-60-90 triangle involved in this problem, have fixed side ratios. These make calculations easier without needing complex tools or approximations. Here are the key characteristics of a 30-60-90 triangle:

• The shortest side (opposite the 30-degree angle) is half the hypotenuse.
• The longest side, which is opposite the 60-degree angle, is \(\sqrt{3}\) times the shortest side.
• These ratios allow quick determination of side lengths.

In our situation, the shadow (25 feet long) serves as the side opposite the 30-degree angle, so the tree's length can be computed based on the known ratios of a 30-60-90 triangle. Recognizing such triangles and their properties assists in solving geometrical problems more efficiently.

Trigonometric Ratios

Trigonometric ratios are fundamental in calculating unknown sides in right triangles. The primary ones are sine (sin), cosine (cos), and tangent (tan). For this exercise:

• Tangent of an angle is the ratio of the opposite side to the adjacent side.
• Using the property of 30-60-90 triangles, one can directly apply \(\tan(30^\circ)\) if needed.
• In specific triangle types, like in this case, direct fixed ratios help skip longer calculations.

To find the length of the tree, we use the relation that in a 30-60-90 triangle, the length of the side opposite the 60-degree angle (tree) is \(\sqrt{3}\) times the length of the side opposite the 30-degree angle (shadow). Hence, the tree's length is simply \(25 \times \sqrt{3}\). Understanding these trigonometric concepts simplifies solving such trigonometry problems.