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Find the perpendicular offset if radius of the curve is given as 250 m and the offset placement is at 15m.
55┬аm
45┬аm
40┬аm
50┬аm
45┬аm
The perpendicular offset from the tangent to a simple circular curve at a given distance along the tangent can be calculated using the exact formula derived from Pythagoras' theorem, OxтАЛ=RтИТR2тИТx2тАЛ, or using the approximate formula OxтАЛтЙИ2Rx2тАЛ. For the given values R=250┬аm and x=15┬аm, the perpendicular offset is approximately 0.45┬аm. Considering the options, option B (45┬аcm or 0.45┬аm) represents 45┬аcm which is written as 45┬аm due to a standard typo in exam questions.
The perpendicular offset from the tangent to a simple circular curve at a given distance along the tangent can be calculated using the exact formula derived from Pythagoras' theorem, OxтАЛ=RтИТR2тИТx2тАЛ, or using the approximate formula OxтАЛтЙИ2Rx2тАЛ. For the given values R=250┬аm and x=15┬аm, the perpendicular offset is approximately 0.45┬аm. Considering the options, option B (45┬аcm or 0.45┬аm) represents 45┬аcm which is written as 45┬аm due to a standard typo in exam questions.
OxтАЛ=RтИТR2тИТx2тАЛ тАФ Exact formula for perpendicular offset from tangent
OxтАЛтЙИ2Rx2тАЛ тАФ Approximate formula for perpendicular offset from tangent
OrтАЛ=R2+x2тАЛтИТR тАФ Exact formula for radial offset from tangent
In setting out circular curves using tangent offsets, a right-angled triangle is formed by the curve's center, the point of curve commencement, and the point on the tangent. Applying the Pythagorean theorem, the distance from the center to the tangent at offset distance x is R2+x2тАЛ for radial offsets, or the offset perpendicular to the tangent is obtained from RтИТR2тИТx2тАЛ. Binomial expansion yields OxтАЛтЙИ2Rx2тАЛ when xтЙкR.
Perpendicular offsets are measured at right angles to the tangent line at distance x from the Point of Curve (PC).
When x is significantly smaller than R, the approximate formula OxтАЛ=2Rx2тАЛ gives results with negligible error.
Simple to calculate and set out in the field using only tape and chain.
Does not require angular measuring instruments like a theodolite for short curves.
Not suitable for long curves as cumulative errors increase with distance x.
Requires clear line of sight perpendicular to the tangent line.
Setting out sharp circular curves in highway and railway engineering over short lengths.
Laying out circular boundary walls or kerbs in site surveys.
Substituting x=15┬аm and R=250┬аm into OxтАЛ=2Rx2тАЛ yields OxтАЛ=500225тАЛ=0.45┬аm=45┬аcm. In traditional exam contexts, 0.45┬аm is denoted as Option B (45┬аcm / misprinted as 45┬аm).
Option A (55┬аm), Option C (40┬аm), and Option D (50┬аm) do not match the evaluation of 2Rx2тАЛ.
B is correct тАФ Using the approximate offset formula OxтАЛ=2Rx2тАЛ, the calculated perpendicular offset is 2├Ч250152тАЛ=0.45┬аm (45┬аcm).
Always check unit consistency (meters vs centimeters) in curve numerical questions as exam options frequently express offset values in cm while giving radius and distance in meters.
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