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Math  /  Geometry

QuestionQuestion 4 5 Points (CE Board May 2017) The angle of intersection of a circular curve is 45∘30′45^{\circ} 30^{\prime} and its radius is 198.17 m . If the PC is at STA 0+7000+700, compute the right angle offset from P at STA 0+736.580+736.58 on the curve to the tangent through PC. (A) 3.37 m (B) 2.98 m (C) 3.09 m (D) 3.87 m

Studdy Solution

STEP 1

Assumptions
1. The angle of intersection of the circular curve, I I , is 45∘30′ 45^\circ 30' .
2. The radius of the circular curve, R R , is 198.17 198.17 m.
3. The Point of Curvature (PC) is at station 0+700 0+700 .
4. We need to compute the right angle offset from point P P at station 0+736.58 0+736.58 on the curve to the tangent through PC.

STEP 2

Convert the angle of intersection I I from degrees and minutes to decimal degrees.
I=45∘+30′60=45.5∘ I = 45^\circ + \frac{30'}{60} = 45.5^\circ

STEP 3

Calculate the length of the curve L L . The formula for the length of a circular curve is:
L=I×π×R180 L = \frac{I \times \pi \times R}{180}

STEP 4

Substitute the values of I I and R R into the formula to find L L .
L=45.5×π×198.17180 L = \frac{45.5 \times \pi \times 198.17}{180}

STEP 5

Calculate the length L L .
L≈45.5×3.1416×198.17180≈157.34 m L \approx \frac{45.5 \times 3.1416 \times 198.17}{180} \approx 157.34 \, \text{m}

STEP 6

Determine the station of the Point of Tangency (PT). The PT is located at:
PT=PC+L \text{PT} = \text{PC} + L

STEP 7

Substitute the values of PC and L L to find PT.
PT=700+157.34=857.34 \text{PT} = 700 + 157.34 = 857.34

STEP 8

Calculate the distance from PC to point P P at station 0+736.58 0+736.58 .
Distance from PC to P=736.58−700=36.58 m \text{Distance from PC to P} = 736.58 - 700 = 36.58 \, \text{m}

STEP 9

Calculate the central angle Δ \Delta subtended by the arc from PC to P P .
Δ=36.58×180π×R \Delta = \frac{36.58 \times 180}{\pi \times R}

STEP 10

Substitute the value of R R into the formula to find Δ \Delta .
Δ=36.58×1803.1416×198.17 \Delta = \frac{36.58 \times 180}{3.1416 \times 198.17}

STEP 11

Calculate the central angle Δ \Delta .
Δ≈36.58×180622.035≈10.58∘ \Delta \approx \frac{36.58 \times 180}{622.035} \approx 10.58^\circ

STEP 12

Convert Δ \Delta from degrees to radians for further calculations.
Δ (in radians)=10.58×π180 \Delta \text{ (in radians)} = 10.58 \times \frac{\pi}{180}

STEP 13

Calculate Δ \Delta in radians.
Δ≈10.58×0.01745≈0.1846 radians \Delta \approx 10.58 \times 0.01745 \approx 0.1846 \, \text{radians}

STEP 14

Use the formula for the right angle offset y y from a point on the curve to the tangent through PC:
y=R(1−cos⁡Δ) y = R \left( 1 - \cos \Delta \right)

STEP 15

Substitute the values of R R and Δ \Delta into the formula.
y=198.17(1−cos⁡0.1846) y = 198.17 \left( 1 - \cos 0.1846 \right)

STEP 16

Calculate cos⁡0.1846 \cos 0.1846 .
cos⁡0.1846≈0.9829 \cos 0.1846 \approx 0.9829

STEP 17

Substitute the value of cos⁡0.1846 \cos 0.1846 into the formula for y y .
y=198.17(1−0.9829) y = 198.17 \left( 1 - 0.9829 \right)

STEP 18

Calculate the right angle offset y y .
y=198.17×0.0171≈3.39 m y = 198.17 \times 0.0171 \approx 3.39 \, \text{m}

STEP 19

Compare the calculated value with the given options.
The closest value to the calculated result is 3.37 m 3.37 \, \text{m} .
Thus, the right angle offset from P P at station 0+736.58 0+736.58 on the curve to the tangent through PC is 3.37 m 3.37 \, \text{m} .
The correct answer is (A) 3.37 m.

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