A 2 m tall boy is 30 m away from a tower. The angle of elevation of the top of the tower from his eye is 45°. What is the height of the tower?

32 m

The correct answer is Option (3) → 32 m

1. Identify the Given Information

• Height of the boy ($h$): $2\text{ m}$
• Distance from the tower ($d$): $30\text{ m}$
• Angle of elevation ($\theta$): $45^\circ$

2. Set Up the Calculation

The boy's eye is $2\text{ m}$ above the ground. The angle of elevation is measured from his eye to the top of the tower. Let $x$ be the height of the tower above the boy's eye level.

In the right-angled triangle:

$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

$\tan(45^\circ) = \frac{x}{30}$

Since $\tan(45^\circ) = 1$:

$1 = \frac{x}{30}$

$x = 30\text{ m}$

3. Calculate Total Height

The total height of the tower ($H$) is the sum of the height above the eye level ($x$) and the height of the boy ($h$):

$H = x + h$

$H = 30\text{ m} + 2\text{ m}$

$H = 32\text{ m}$

Conclusion

The total height of the tower is $32\text{ m}$.