The Ramp Angle Decision
A city is building a wheelchair ramp to a community centre entrance that is 1.2 m above ground. Accessibility guidelines require the ramp angle to be no more than 4.8°. The available horizontal run is 14 m. An engineer uses the sine ratio to check the angle: sin θ = 1.2/14, giving θ ≈ 4.9°. A contractor argues that the tangent ratio should be used instead: tan θ = 1.2/14, giving θ ≈ 4.9° as well, so 'it doesn't matter.' A second engineer says the difference between sine and tangent matters at larger angles but not here. The city must decide whether the ramp meets the guideline and which ratio is actually correct.
1. Determine which trigonometric ratio correctly models the ramp angle and whether the ramp meets the 4.8° guideline.
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The ramp forms a right triangle where:
- the vertical rise (1.2 m) is the side opposite the angle at the base,
- the horizontal run (14 m) is the side adjacent to the angle at the base,
- the ramp length is the hypotenuse.
The correct ratio relating opposite and adjacent is tangent: tan θ = opposite/adjacent = 1.2/14.
θ = arctan(1.2/14) ≈ arctan(0.08571) ≈ 4.90°.
Since 4.90° > 4.8°, the ramp does not meet the guideline.
The engineer who used sine was wrong in principle: sin θ = opposite/hypotenuse, and 14 m is the run, not the hypotenuse. Using sine with 14 as hypotenuse gives θ ≈ 4.92°, which is numerically close but conceptually incorrect. The contractor's claim that 'it doesn't matter' is false in general; the ratios differ, and at small angles they happen to be close because sin θ ≈ tan θ for small θ.
2. The contractor says 'sine and tangent give almost the same answer, so it doesn't matter.' Explain why this reasoning is dangerous even if the numerical difference is small here.
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The reasoning is dangerous because it treats a coincidence as a general rule. For small angles, sin θ ≈ tan θ ≈ θ (in radians), so the numerical answers are close. But this approximation breaks down as the angle grows.
For example, at θ = 30°: sin 30° = 0.5, tan 30° ≈ 0.577, a difference of about 15%. At θ = 45°: sin 45° ≈ 0.707, tan 45° = 1, a difference of about 41%. Using the wrong ratio could produce a design that fails safety requirements.
The correct reasoning is to identify the sides relative to the angle and choose the ratio that matches those sides. Here, opposite and adjacent are known, so tangent is correct. The fact that the wrong ratio gives a similar number is not a justification; it is an accident of the small angle.
3. If the city wants the ramp to meet the 4.8° guideline, what is the minimum horizontal run required for a 1.2 m rise? Round to two decimal places.
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Using tan θ = opposite/adjacent, with θ = 4.8° and opposite = 1.2 m:
tan(4.8°) = 1.2 / run
run = 1.2 / tan(4.8°).
tan(4.8°) ≈ 0.08397.
run ≈ 1.2 / 0.08397 ≈ 14.29 m.
The minimum run is approximately 14.29 m. The available 14 m is too short, confirming the ramp does not meet the guideline. The city would need to extend the run by about 0.29 m, or reduce the rise.