Solving Complex Height and Distance Problems

🔄 Quick Recap

In our previous section, we solved basic height and distance problems using a single angle and one unknown. Now, we'll tackle more challenging problems that involve multiple angles or multiple unknowns.

📚 Types of Complex Problems

In real-life situations, problems often involve more complexity than our basic examples. Here are some common types:

1. Two angles of elevation to the same object - useful when the object is very tall
2. Object on top of another object - like a flagpole on a building
3. Problems involving both angles of elevation and depression
4. Moving objects - where angles change over time

🖼️ Visual Aids: Object on Top of Another Object

Consider a flagpole mounted on top of a building:

In this situation, we need to find both the height of the building and the length of the flagpole by using different angles of elevation.

✅ Solved Example 1: Flagpole on a Building

Problem: From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building, and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from point P.

Solution:

Step 1: Understand the problem

• The building height (h₁) = 10 m
• Angle of elevation to top of building = 30°
• Angle of elevation to top of flagstaff = 45°
• We need to find:
  • The length of the flagstaff (h₂)
  • The distance from point P to the building (d)

Step 2: Find the distance d using the information about the building

tan 30° = 10/d
d = 10/tan 30°
d = 10/(1/√3)
d = 10 × √3
d = 10 × 1.732
d = 17.32 m

Step 3: Let's call the flagstaff length x meters

• Then the total height (building + flagstaff) = 10 + x meters

Step 4: Use the 45° angle to form another equation

tan 45° = (10 + x)/d
1 = (10 + x)/17.32
17.32 = 10 + x
x = 7.32 m

Therefore:

• The length of the flagstaff is 7.32 m
• The distance from point P to the building is 17.32 m

🖼️ Visual Aids: Bridge Over a River

Here's a problem involving angles of depression from a bridge:

✅ Solved Example 2: River Width

Problem: From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

Solution:

Step 1: Understand the problem

• Height of the bridge above the banks (h) = 3 m
• Angle of depression to one bank = 30°
• Angle of depression to the other bank = 45°
• We need to find the width of the river

Step 2: Find the distance from the bridge to the first bank (d₁)

tan 30° = 3/d₁
d₁ = 3/tan 30°
d₁ = 3/(1/√3)
d₁ = 3 × √3
d₁ = 5.2 m (approximately)

Step 3: Find the distance from the bridge to the second bank (d₂)

tan 45° = 3/d₂
d₂ = 3/tan 45°
d₂ = 3/1
d₂ = 3 m

Step 4: Calculate the total width of the river

Width = d₁ + d₂ = 5.2 + 3 = 8.2 m

Therefore, the width of the river is approximately 8.2 m.

✅ Solved Example 3: Broken Tree

Problem: A tree breaks due to a storm, and the broken part bends so that the top of the tree touches the ground, making an angle of 30° with it. The distance between the foot of the tree and the point where the top touches the ground is 8 m. Find the height of the tree.

Solution:

Step 1: Understand the problem

• The tree broke and its top touched the ground
• The broken part makes an angle of 30° with the ground
• Distance from tree base to where top touches ground = 8 m
• We need to find the original height of the tree

Step 2: Let's call the height of the tree h meters

• The distance from the foot of the tree to the point where the top touches the ground is 8 m
• At the breaking point, the tree forms a right-angled triangle

Step 3: Use the tangent ratio to find the height of the unbroken part

tan 30° = opposite/adjacent
tan 30° = y/8 (where y is the height of the unbroken part)
y = 8 × tan 30°
y = 8 × (1/√3)
y = 8 × 0.5774
y = 4.62 m

Step 4: The broken part forms a hypotenuse of a right-angled triangle

• Let's call the length of the broken part z meters
• By Pythagorean theorem:

z² = 8² + y²
z² = 64 + 21.34
z² = 85.34
z = 9.24 m

Step 5: Calculate the total height of the tree

Total height of tree = y + z = 4.62 + 9.24 = 13.86 m

Therefore, the original height of the tree was approximately 13.86 m.

⚖️ Quick Comparison/Summary Table

Problem Type	Key Characteristics	Approach
Single object, one angle	One angle of elevation/depression, one unknown	Use basic trigonometric ratios directly
Object on another object	Multiple angles to different parts	Solve step by step, using one angle first
Objects on opposite sides	Angles on opposite sides	Find distances separately, then combine
Moving objects	Changing angles over time	Use angles at different times to create equations

🧠 Memory Tricks

When solving complex problems:

• Always Draw a diagram (D)
• Identify what you know and what you need to find (I)
• Generate equations using trigonometric ratios (G)
• Solve step by step (S)

Remember DIGS to dig out the solution!

⚠️ Common Misconceptions

1. Misconception: We need to solve for all unknowns at once. Correction: Break the problem into steps, solving for one unknown at a time.

2. Misconception: The same trigonometric ratio must be used throughout. Correction: Choose the most appropriate ratio for each step of the problem.

3. Misconception: All problems require the Pythagorean theorem. Correction: While it's useful in some cases, many problems can be solved directly with trigonometric ratios.

💡 Key Points to Remember

• Draw clear diagrams with all known information labeled.
• Break complex problems into simpler parts.
• Work step by step, using the result from one step to solve the next.
• Verify your answer makes sense in the context of the problem.
• Always include the appropriate units in your answer.

🤔 Think About It!

1. If you're standing on a cliff and observe two boats in the sea with angles of depression of 30° and 45°, which boat is closer to the cliff?

2. How would the calculation change if the observer's height is given and needs to be considered?

🌍 Real-Life Applications

These complex trigonometry problems have numerous real-world applications:

1. Engineering: Designing tall structures like transmission towers and skyscrapers
2. Military: Range finding and targeting systems
3. Navigation: Maritime navigation and aircraft landing systems
4. Forestry: Estimating tree heights and timber volumes
5. Astronomy: Calculating distances to celestial bodies

🔜 What Next?

Now that we've learned to solve various types of height and distance problems, we'll practice with more real-world examples and prepare for the exercises at the end of the chapter.