A flagpole has a shadow measuring 10 meters. Given that the height of the flagpole is 10√3 meters, What is the angle at which the sun is elevated?

Correct — C. The pole, its shadow and the sun's ray form a right triangle. The angle of elevation θ sits at the tip of the shadow.

tan θ = opposite ⁄ adjacent = height ⁄ shadow
Substitute: 10√3 ⁄ 10 = √3
tan 60° = √3, so θ = 60° → option (c)

Why the others are wrong

• (a)
  30° — tan 30° = 1⁄√3, which is shadow ⁄ height, the ratio upside down. At 30° a 10 m shadow belongs to a pole 10⁄√3 ≈ 5.77 m tall.
• (b)
  45° — tan 45° = 1 means the height equals the shadow, 10 m. This pole is 10√3 ≈ 17.32 m, taller than its shadow.
• (d)
  75° — tan 75° = 2 + √3 ≈ 3.73, so a 10 m shadow at 75° means a pole about 37.3 m tall, not 10√3 ≈ 17.32 m.

In a height-and-shadow problem the object is the side opposite the sun's angle and the shadow is the side next to it. So tan θ = height ⁄ shadow.

A taller object for the same shadow means a higher sun. Height equal to the shadow gives 45°. Height √3 × the shadow gives 60°, and height 1⁄√3 × the shadow gives 30°.

Key facts

• tan 30° = 1⁄√3, tan 45° = 1, tan 60° = √3.
• Angle of elevation: tan θ = height ⁄ horizontal distance.
• A 30°-60°-90° triangle has sides in the ratio 1 : √3 : 2.

Common traps

• Inverting the ratio to shadow ⁄ height, which gives 1⁄√3 and the wrong answer 30°.
• Reaching for sin or cos when the two known sides are the legs. Height and shadow are both legs, so the ratio is tan.

No directly related past PYQ was found.