A tower generates a shadow that is 40 meters long when the sun’s angle of elevation is 45 degrees. What is the height of the tower?

Correct — B. The tower, its shadow and the sun's ray form a right triangle, with the 45° angle at the tip of the shadow.

tan 45° = height ÷ shadow
Substitute: 1 = h ÷ 40
Multiply: h = 40 × 1 = 40 m → option (b)

Why the others are wrong

• (a)
  50 m — 50 m gives tan θ = 50⁄40 = 1.25, an elevation of about 51°. At 45° the height and the shadow must be equal.
• (c)
  20 m — 20 m gives tan θ = 20⁄40 = 1⁄2, an elevation of about 27°. A tower half as tall as its shadow needs a lower sun than 45°.
• (d)
  10 m — 10 m is a quarter of the shadow: tan θ = 1⁄4, an elevation near 14°. At 45°, tan θ is exactly 1, so h = 40 m.

In heights-and-distances questions the object, its shadow and the sun's ray make a right triangle, with the angle of elevation at the tip of the shadow.

tan θ = opposite ÷ adjacent = height ÷ shadow.

At 45°, tan θ = 1, so the triangle is isosceles and the height equals the shadow. At 60° the height is √3 times the shadow. At 30° it is the shadow ÷ √3.

Key facts

• tan 30° = 1⁄√3, tan 45° = 1 and tan 60° = √3.
• At a 45° angle of elevation, an object's height equals the length of its shadow.

Common traps

• Using sin 45° instead of tan 45°, which gives 40 × 1⁄√2 ≈ 28.3 m.
• Swapping height and shadow in tan θ, which goes unpunished at 45° (tan = cot = 1) but fails at 30° and 60°.

No directly related past PYQ was found.