Problem 9 - A Complete Resource Book in Mathematics for JEE Main

Question

A person standing at the foot of a tower walks a distance \(3 a\) away from the tower and observes that the angle of elevation of the top of the tower is \(\alpha\). He then walks a distance \(4 a\) perpendicular to the previous direction and observes the angle of elevation to be \(\beta\). Then height of the tower is (A) \(3 a \tan \alpha\) or \(5 a \tan \beta\) (B) \(5 a \tan \alpha\) or \(3 a \tan \beta\) (C) \(4 a \tan \beta\) (D) \(7 a \tan \alpha\)

Step-by-Step Solution

Answer

(A) \(3a \tan \alpha\) or \(5a \tan \beta\).

Step 1: Understanding the problem

We have a tower and a person observing it from two different points. Initially, the person walks a distance \(3a\) from the base, making an angle of elevation \(\alpha\) with the top of the tower. The person then walks \(4a\) perpendicular to the first direction and observes the tower at an angle \(\beta\). We need to find the height of the tower.

Step 2: Setting up the scenario

Let's denote the height of the tower as \(h\). Initially, the observer stands at a distance of \(3a\) from the base of the tower. Hence, the right triangle is formed with the height \(h\) and base \(3a\).

Step 3: Using the tangent function for the first position

The angle of elevation from the first position is \(\alpha\). Therefore, we have the equation:\[ \tan \alpha = \frac{h}{3a} \]Solving for \(h\), we get:\[ h = 3a \tan \alpha \] (Equation 1)

Step 4: Setting up the second scenario

From the second position, the observer has moved \(4a\) perpendicular to the initial position, which forms another right triangle with base \(5a\) (calculated using the Pythagorean theorem: \((3a)^2 + (4a)^2 = (5a)^2)\) for the complete distance from the tower.

Step 5: Using the tangent function for the second position

Now, from the second position, the angle of elevation is \(\beta\). Hence, we can write:\[ \tan \beta = \frac{h}{5a} \]Solving for \(h\), we get:\[ h = 5a \tan \beta \] (Equation 2)

Step 6: Comparing the results

Combining Equation 1 and Equation 2, we can see the height of the tower \(h\) can either be expressed as \(3a \tan \alpha\) or \(5a \tan \beta\).

Key Concepts

Angle of Elevation

Imagine you are looking up at the top of a tower from where you stand. The angle that your line of sight makes with the horizontal ground is known as the angle of elevation. It's like tilting your head upwards to look at the top of the tower.

• This angle helps us determine the height of tall objects without needing to climb or measure it directly.

In trigonometry, the angle of elevation is crucial because it allows us to use various trigonometric functions, such as tangent, to find unknown distances or heights in right triangles.

Tangent Function

The tangent function comes from a trigonometric ratio that involves the angle of elevation in a right triangle setup, like the one we find when observing a tower. In simplest terms, it is the ratio of the opposite side to the adjacent side of the angle of elevation:

• Formula: \[ \tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \]

For the first scenario described in our exercise, the tangent of angle \(\alpha\) gives us the height of the tower compared to the 3a distance to the base. Similarly, the second scenario involves the tangent of angle \(\beta\) using the full base distance calculated with the Pythagorean theorem, leading to a base value of 5a.

Right Triangle Properties

A right triangle has one 90-degree angle, and it forms the foundation for many trigonometric solutions. This kind of triangle allows us to employ various mathematical calculations to understand relationships between angles and lengths.

• Important properties include the Pythagorean theorem: \((\text{Base})^2 + (\text{Other Side})^2 = (\text{Hypotenuse})^2\).

In this exercise, the observer at two varying positions makes two separate right triangles. The first triangle uses the distance 3a, and the second triangle's distance of 4a is adopted and adjusted using the Pythagorean theorem to reach a combined distance or base of 5a.