What is a Problem on Two Towers? | Simple Explanation for Class 10

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What is an Example of a Problem on Two Towers?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A 'Problem on Two Towers' is a classic geometry problem where you need to find distances or heights involving two vertical structures (like towers, poles, or buildings) and a point on the ground between them or outside them. It often uses concepts like trigonometry (angles of elevation/depression) and Pythagoras theorem to solve for unknown values.

Simple Example

Quick Example

Imagine two mobile network towers standing tall. You are standing exactly between them. If you know how tall each tower is and how far you are from the base of one tower, a 'Problem on Two Towers' would ask you to find the total distance between the two towers, or the angle you need to look up to see the top of the other tower.

Worked Example

Step-by-Step

Two towers, Tower A and Tower B, are 60 meters apart. Tower A is 30 meters tall. From the top of Tower A, the angle of depression to the base of Tower B is 30 degrees. Find the height of Tower B.

---Step 1: Draw a diagram. Draw two vertical lines for the towers and a horizontal line connecting their bases. Draw a line from the top of Tower A horizontally towards Tower B. Then draw a line from the top of Tower A to the base of Tower B. This forms a right-angled triangle.

---Step 2: Identify knowns. Distance between towers = 60m. Height of Tower A = 30m. Angle of depression from top of Tower A to base of Tower B = 30 degrees. This means the angle inside the triangle at the top of Tower A (between the horizontal line and the line to Tower B's base) is 30 degrees.

---Step 3: Use trigonometry. In the right-angled triangle formed, the horizontal distance is 60m (adjacent side to the 30-degree angle). Let 'h' be the vertical distance from the top of Tower A's horizontal line down to the base of Tower B. We use tan(angle) = opposite/adjacent. So, tan(30 degrees) = h / 60.

---Step 4: Calculate 'h'. We know tan(30 degrees) is approximately 1/sqrt(3) or 0.577. So, h = 60 * tan(30 degrees) = 60 * 0.577 = 34.62 meters.

---Step 5: Find the height of Tower B. The height 'h' is the distance from the horizontal line (at 30m height) down to the base of Tower B. So, the total height of Tower B is the height of Tower A plus 'h'. Height of Tower B = 30m + 34.62m = 64.62 meters.

Answer: The height of Tower B is approximately 64.62 meters.

Why It Matters

Understanding 'Problems on Two Towers' is crucial for fields like engineering, architecture, and even space technology. Civil engineers use these principles to design stable buildings and bridges, ensuring they can withstand forces. Architects apply these concepts to calculate line of sight and structural integrity, making sure their designs are both safe and beautiful.

Common Mistakes

MISTAKE: Confusing angles of elevation and depression. Students often use the angle given directly inside the triangle without considering if it's from the horizontal or vertical. | CORRECTION: Always draw a clear diagram. Remember, the angle of depression is from a horizontal line looking DOWN, and the angle of elevation is from a horizontal line looking UP.

MISTAKE: Incorrectly identifying the opposite, adjacent, and hypotenuse sides relative to the angle. | CORRECTION: For the chosen angle, the side directly across is 'opposite', the side next to it (not the hypotenuse) is 'adjacent', and the longest side opposite the 90-degree angle is 'hypotenuse'. Use SOH CAH TOA.

MISTAKE: Not considering all parts of the height. If one tower is taller than the other, the calculation might only give the difference in height, not the full height of the second tower. | CORRECTION: After finding the missing segment, always add or subtract it from the known height to get the final total height or distance.

Practice Questions

Try It Yourself

QUESTION: A 15-meter tall pole casts a shadow. From the top of the pole, the angle of depression to the tip of its shadow is 45 degrees. How long is the shadow? | ANSWER: 15 meters

QUESTION: Two buildings are 40 meters apart. From the top of the shorter building, which is 25 meters tall, the angle of elevation to the top of the taller building is 30 degrees. What is the height of the taller building? (Use tan(30) = 0.577) | ANSWER: Approximately 48.08 meters

QUESTION: From a point P on the ground between two poles, the angles of elevation to the tops of the poles are 60 degrees and 30 degrees respectively. If the height of the first pole is 20 meters and the distance of point P from its base is 20/sqrt(3) meters, find the height of the second pole and the total distance between the poles. (Use sqrt(3) = 1.732) | ANSWER: Height of second pole = 20/3 meters (approx 6.67m); Total distance between poles = 20/sqrt(3) + 20/sqrt(3) = 40/sqrt(3) meters (approx 23.09m)

MCQ

Quick Quiz

From the top of a 50m tall lighthouse, the angle of depression to a boat is 45 degrees. How far is the boat from the base of the lighthouse?

25 meters

50 meters

50 * sqrt(2) meters

100 meters

The Correct Answer Is:

When the angle of depression is 45 degrees, the triangle formed is an isosceles right-angled triangle. This means the opposite side (height of lighthouse) is equal to the adjacent side (distance to the boat). So, if the lighthouse is 50m tall, the boat is 50m away.

Real World Connection

In the Real World

These principles are used by surveyors and construction workers in India. For example, when building a new flyover or a metro line in a city like Delhi or Mumbai, engineers use trigonometry to calculate the precise heights and distances needed for pillars and bridges, ensuring everything aligns perfectly and is safe for daily use.

Key Vocabulary

Key Terms

ANGLE OF ELEVATION: The angle measured upwards from the horizontal line of sight | ANGLE OF DEPRESSION: The angle measured downwards from the horizontal line of sight | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle | TRIGONOMETRY: A branch of mathematics dealing with the relationships between the sides and angles of triangles | SOH CAH TOA: A mnemonic for the definitions of the three basic trigonometric functions: Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent

What's Next

What to Learn Next

Great job with 'Problems on Two Towers'! Next, you can explore 'Heights and Distances with Multiple Angles'. This will involve solving more complex scenarios with two or more angles of elevation or depression, often requiring you to set up and solve systems of equations, which is a super useful skill!