The angle of elevation of the top of a tower from a point on the ground which is 30m away from the foot of tower is $30^{\circ}$. What is the height of tower?

Question

Accepted Answer

The correct option is 2:
To solve this problem, we need to use the concept of trigonometry, specifically the tangent function, which relates the angle of elevation to the height and base of a right-angled triangle.

First, let's visualize the scenario:
Imagine a tower standing vertically on the ground. Let the height of the tower be $h$.
A point on the ground is 30m away from the foot of the tower. This distance represents the base of the right-angled triangle.
The angle of elevation of the top of the tower from this point is $30^{\circ}$. This is the angle between the horizontal ground and the line of sight to the top of the tower.

In a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
In our case:
Opposite side = Height of the tower ($h$)
Adjacent side = Distance from the foot of the tower = 30 m
Angle of elevation = $30^{\circ}$

So, we can write the trigonometric relationship as:
$$\tan(\text{angle of elevation}) = \frac{\text{Opposite}}{\text{Adjacent}}$$
$$\tan(30^{\circ}) = \frac{h}{30}$$

We know the value of $\tan(30^{\circ})$ is $\frac{1}{\sqrt{3}}$.
Substitute this value into the equation:
$$\frac{1}{\sqrt{3}} = \frac{h}{30}$$

Now, we need to solve for $h$. Multiply both sides of the equation by 30:
$$h = \frac{30}{\sqrt{3}}$$

To rationalize the denominator, multiply the numerator and the denominator by $\sqrt{3}$:
$$h = \frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$$
$$h = \frac{30\sqrt{3}}{3}$$
$$h = 10\sqrt{3}\text{ m}$$

Thus, the height of the tower is $10\sqrt{3}$ m.

The incorrect option is 1: $30\sqrt{3}m$ is incorrect because it implies that the adjacent side was multiplied by $\tan(30^{\circ})$ as $30 \times \sqrt{3}$ or that $\cot(30^{\circ})$ was mistakenly used as $h = 30 \cot(30^{\circ})$, leading to $30\sqrt{3}$.
The incorrect option is 3: 30 m is incorrect because this is the distance from the foot of the tower, not the height. This would only be true if the angle of elevation was $45^{\circ}$.
The incorrect option is 4: $15\sqrt{3}m$ is incorrect, possibly resulting from an arithmetic error during simplification or an incorrect trigonometric ratio. For example, if one mistakenly used $h = \frac{30 \times \sqrt{3}}{2}$, which is not derived from the correct trigonometric relationship.