Question

At some time of the day, the length of the shadow of a tower is equal to its height, then the sun's attitude at that time is :

• A. \(30^\circ\)
• B. \(60^\circ\)
• C. \(90^\circ\)
• D. \(45^\circ\)

Hint:

1. Draw a right-angled triangle:
Vertical side = Height of tower (\(h\)).
Horizontal side = Length of shadow (\(s\)).
Angle of elevation of the sun = \(\theta\). 2. Use trigonometry: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{s}\). 3. Given condition: Length of shadow = Height of tower, so \(s = h\). 4. Substitute: \(\tan \theta = \frac{h}{h} = 1\). 5. Find \(\theta\): If \(\tan \theta = 1\), then \(\theta = 45^\circ\). (This occurs in an isosceles right-angled triangle where the two non-hypotenuse sides are equal).

Answer 1

The Correct Option is D

Concept: This problem involves basic trigonometry, specifically the tangent function, in a right-angled triangle formed by the tower, its shadow, and the line of sight to the sun. The sun's altitude is the angle of elevation of the sun. Step 1: Visualize the scenario Imagine a vertical tower. The sun's rays cast a shadow of the tower on the ground.
Let \(h\) be the height of the tower.
Let \(s\) be the length of the shadow.
Let \(\theta\) be the sun's altitude (angle of elevation from the tip of the shadow to the top of the tower). These three form a right-angled triangle, where the tower is the perpendicular side, the shadow is the base, and \(\theta\) is the angle opposite the tower and adjacent to the shadow. Step 2: Relate height, shadow, and angle using trigonometry In the right-angled triangle formed: \[ \tan \theta = \frac{\text{Opposite side (height of tower)}}{\text{Adjacent side (length of shadow)}} \] \[ \tan \theta = \frac{h}{s} \] Step 3: Apply the given condition The problem states that "the length of the shadow of a tower is equal to its height." So, \(s = h\). Substitute this condition into the tangent equation: \[ \tan \theta = \frac{h}{h} \] Since \(h\) is a height, \(h \neq 0\), so we can simplify: \[ \tan \theta = 1 \] Step 4: Find the angle \(\theta\) We need to find the angle \(\theta\) whose tangent is 1. We know from standard trigonometric values that \(\tan 45^\circ = 1\). Therefore, \(\theta = 45^\circ\). The sun's altitude (angle of elevation) at that time is \(45^\circ\). This matches option (4).