A wheelchair ramp needs to reach a platform that is 1.5 meters high. If the ramp is designed to make an angle of 30° with the level ground, what is the length of the ramp?

Correct — D.
The ramp is the hypotenuse, and the 1.5 m rise is the side opposite the 30° angle.
sin 30° = opposite ÷ hypotenuse
1⁄2 = 1.5 ÷ L

Cross-multiply: L = 1.5 × 2 = 3 m
Check: 3 × sin 30° = 3 × 1⁄2 = 1.5 m, the platform height → option (d)

Why the others are wrong

• (a)
  1.5m — 1.5 m is the height itself. A ramp as long as its rise would need sin θ = 1, a vertical 90° climb. At 30° the ramp must be longer than the rise.
• (b)
  2m — A 2 m ramp gives sin θ = 1.5 ÷ 2 = 0.75, an angle of about 49°. A shorter ramp to the same height is steeper, so 2 m is too steep for a 30° design.
• (c)
  2.5m — A 2.5 m ramp gives sin θ = 1.5 ÷ 2.5 = 0.6, about 37°, still steeper than 30°. A 30° ramp needs the rise to be exactly half its length.

Pick the ratio that links what you know to what you want. Here you know the side opposite the angle (the 1.5 m rise) and want the hypotenuse (the ramp), so the ratio is sin.

sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.

A 30°–60°–90° triangle has sides in the ratio 1 : √3 : 2, so the side opposite 30° is half the hypotenuse. So the ramp is 1.5 × 2 = 3 m.

Key facts

• sin 30° = 1⁄2, cos 30° = √3⁄2, tan 30° = 1⁄√3
• In a 30°–60°–90° triangle the sides are in the ratio 1 : √3 : 2, the shortest side opposite 30°
• The ramp's horizontal run here is 1.5 × √3 ≈ 2.6 m, shorter than the ramp itself

Common traps

• Using tan 30°, which gives the horizontal run 1.5√3 ≈ 2.6 m instead of the ramp's length
• Writing L = 1.5 × sin 30° = 0.75 m, a hypotenuse shorter than the rise it climbs

No directly related past PYQ was found.