Question

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be \( 45^\circ \). The height (in metres) of the tower is :

• A. \( 10\sqrt{3} \)
• B. \( 30\sqrt{3} \)
• C. \( 60 \)
• D. \( 30 \)

Hint:

Whenever the angle of elevation is \( 45^\circ \), the height is always equal to the distance from the base.

Answer 1

The Correct Option is C

We need to find the height of a vertical tower given a certain angle of elevation from a point on the ground. The question provides the following key information:

• The distance from the point on the ground to the foot of the tower is \( 60 \) meters.
• The angle of elevation from this point to the top of the tower is \( 45^\circ \).

To find the height of the tower, we can use basic trigonometry. Specifically, we apply the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle.

The tangent of an angle in a right triangle is defined as:

\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

Here, \(\theta = 45^\circ\), the opposite side to the angle is the height of the tower \( h \), and the adjacent side is the distance from the point to the base of the tower, \( 60 \) m.

Substituting these into the formula, we get:

\(\tan(45^\circ) = \frac{h}{60}\)

Since \( \tan(45^\circ) = 1 \), we have:

\(1 = \frac{h}{60}\)

Solving for \( h \), we multiply both sides by \( 60 \):

\(h = 60\)

Therefore, the height of the tower is \( 60 \) meters.

This calculation shows why the correct answer is the third option, \( 60 \) meters, confirming that the problem provided all necessary details for this straightforward trigonometric determination.