What is Solving Problems on Heights and Distances?

S6-SA2-0035

Grade Level:

Class 10

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

Definition

What is it?

Solving Problems on Heights and Distances means using trigonometry to find unknown heights or distances of objects, like a tall building or the distance across a river, by measuring angles and one known length. It helps us understand the world around us without physically measuring everything.

Simple Example

Quick Example

Imagine you are standing some distance from a tall water tank in your village. You want to know how tall it is, but you can't climb it with a measuring tape. If you know how far you are standing from its base and you can measure the angle from your eye to the top of the tank, you can use trigonometry to find its height.

Worked Example

Step-by-Step

Let's find the height of a pole if its shadow is 15 meters long and the angle of elevation of the sun is 30 degrees.

1. **Understand the setup:** We have a right-angled triangle. The pole is the 'opposite' side, the shadow is the 'adjacent' side, and the angle of elevation is 30 degrees.
---2. **Choose the correct trigonometric ratio:** Since we know the adjacent side (shadow) and want to find the opposite side (pole's height), we use the 'tan' ratio (tan = opposite/adjacent).
---3. **Set up the equation:** tan(30 degrees) = Height of pole / Length of shadow
---4. **Substitute known values:** tan(30 degrees) = Height of pole / 15 meters
---5. **Recall tan(30 degrees) value:** We know tan(30 degrees) = 1/sqrt(3) or approximately 0.577.
---6. **Solve for the Height of pole:** Height of pole = 15 * tan(30 degrees) = 15 * (1/sqrt(3)) = 15 / 1.732
---7. **Calculate the final answer:** Height of pole approximately = 8.66 meters.

**Answer:** The height of the pole is approximately 8.66 meters.

Why It Matters

This concept is crucial for engineers designing buildings, pilots navigating flights, and even ISRO scientists planning satellite launches. It helps in precise measurements without direct physical contact, making it vital for construction, mapping, and space technology careers.

Common Mistakes

MISTAKE: Confusing angle of elevation with angle of depression. | CORRECTION: Angle of elevation is always measured UP from the horizontal line of sight, while angle of depression is measured DOWN from the horizontal line of sight.

MISTAKE: Using the wrong trigonometric ratio (e.g., using sine instead of tangent). | CORRECTION: Always draw a clear diagram, identify the known and unknown sides relative to the given angle (opposite, adjacent, hypotenuse), and then choose the correct ratio (SOH CAH TOA).

MISTAKE: Not converting units or rounding off too early. | CORRECTION: Ensure all measurements are in the same units before calculation. For intermediate steps, keep more decimal places and only round the final answer to the required precision.

Practice Questions

Try It Yourself

QUESTION: A ladder 10 meters long leans against a wall. If it makes an angle of 60 degrees with the ground, what is the height of the wall that the ladder reaches? | ANSWER: 8.66 meters

QUESTION: From a point on the ground 50 meters away from the foot of a tower, the angle of elevation of the top of the tower is 45 degrees. Find the height of the tower. | ANSWER: 50 meters

QUESTION: Two posts are 100 meters apart. The angle of elevation of the top of the first post from the foot of the second post is 30 degrees. The angle of elevation of the top of the second post from the foot of the first post is 60 degrees. Find the difference in their heights. | ANSWER: Approximately 57.74 meters

MCQ

Quick Quiz

If the angle of elevation of the sun is 30 degrees, and a tree casts a shadow of 20 meters, what is the height of the tree?

10 meters

20/sqrt(3) meters

20 * sqrt(3) meters

40 meters

The Correct Answer Is:

We use tan(angle) = opposite/adjacent. Here, tan(30) = Height / 20. Since tan(30) = 1/sqrt(3), Height = 20/sqrt(3) meters. So option B is correct.

Real World Connection

In the Real World

When a civil engineer designs a flyover or a bridge in a busy city like Bengaluru, they use heights and distances calculations to ensure the structure's stability and correct slope. Even drone operators use similar principles to map areas or plan delivery routes for services like Zepto, calculating altitudes and distances to avoid obstacles.

Key Vocabulary

Key Terms

ANGLE OF ELEVATION: Angle measured upwards from the horizontal line of sight to an object | ANGLE OF DEPRESSION: Angle measured downwards from the horizontal line of sight to an object | LINE OF SIGHT: The imaginary line from the observer's eye to the object being viewed | TRIGONOMETRIC RATIOS: Ratios like sine, cosine, tangent that relate angles and side lengths in a right-angled triangle

What's Next

What to Learn Next

Great job understanding heights and distances! Next, you can explore 'Applications of Trigonometry in Real Life' to see more practical uses, or dive into 'Trigonometric Identities' to learn how to simplify complex trigonometric expressions. Keep practicing!