Problem 1

Given points:

A(N1200, E1850), B(N2100, E2875), C(N1900, E4000).

We treat coordinates as (x=Easting, y=Northing):

A(1850,1200), B(2875,2100), C(4000,1900)

We use the circumcenter formulas:

D = 2[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]

Plug in:

D = 2[1850(2100-1900)+2875(1900-1200)+4000(1200-2100)]

D = 2[1850(200)+2875(700)+4000(-900)]

D = 2[370000 + 2012500 – 3600000]

D = 2[-1217500] = -2435000

Compute Ux:

Ux = (1850^2+1200^2)(2100-1900)+(2875^2+2100^2)(1900-1200)+(4000^2+1900^2)(1200-2100) / D

Compute each piece:

1850^2+1200^2 = 3422500

2875^2+2100^2 = 13015625

4000^2+1900^2 = 19761000

Now plug:

Ux = [3422500(200) + 13015625(700) + 19761000(-900)] / -2435000

Ux = (684500000 + 9110937500 – 17784900000) / -2435000

Ux = -8039475000 / -2435000 = 3204.748

Now Uy:

Uy = [3422500(4000-2875)+13015625(1850-4000)+19761000(2875-1850)] / D

Compute:

4000-2875=1125, 1850-4000=-2150, 2875-1850=1025

Uy = [3422500(1125) + 13015625(-2150) + 19761000(1025)] / -2435000

Uy = (3840312500 – 27983671875 + 20253075000) / -2435000

Uy = -3890284375 / -2435000 = 690.773

Radius using point A:

R = sqrt[(3204.748-1850)^2 + (690.773-1200)^2]

R = sqrt[1354.748^2 + (-509.227)^2]

R = sqrt(1834036 + 259307) = sqrt(2093343)

R = 1447.293 ft

Final Answers:

O(N=690.773, E=3204.748), R = 1447.293 ft

Problem 2

Points:

A(350,450), B(875,850), C(1100,200)

AB

N = 875 – 350 = 525

E = 850 – 450 = 400

AB = sqrt(525^2 + 400^2) = sqrt(275625 + 160000)

AB = sqrt(435625) = 660.019 ft

Bearing:

= tan^{-1}(400/525) = 37.3039

Bearing AB: N 37.3039 E

BC

N = 1100-875 = 225

E = 200-850 = -650

BC = sqrt[225^2 + (-650)^2] = sqrt(50625 + 422500)

BC = sqrt(473125) = 687.841 ft

Bearing:

= tan^{-1}(650/225) = 70.9065

Bearing BC: N 70.9065 W

Problem 3

Given:

PVI = Sta 8+00, Elev = 310.50

g1 = -1.5% = -0.015, g2 = +2.0% = 0.02

L = 7 stations = 700 ft

a = (g2 – g1) / 2L = [0.02 – (-0.015)] / 1400

a = 0.035 / 1400 = 0.000025

BVC station:

BVC = 8+00 – 3.5 = 4+50

Elevation at BVC:

z_BVC = 310.50 – (-0.015)(350) – 0.000025(350^2)

z_BVC = 310.50 + 5.25 – 3.0625 = 312.6875 ft

Elevation at Sta 7+10

x = 710 – 450 = 260 ft

z = 312.6875 + (-0.015)(260) + 0.000025(260^2)

z = 312.6875 – 3.90 + 1.69 = 310.4775

Elevation at 7+10 = 310.478 ft

Lowest Point

x_LP = -g1 / 2a = 0.015 / 0.00005 = 300 ft

Station LP = 450 + 300 = 750 = 7+50

z_LP = 312.6875 – 4.50 + 2.25 = 310.4375

LP Elevation = 310.438 ft

Problem 4

Given:

VPI = Sta 87+25, Elev = 115.78

g1 = -3.5% = -0.035

g2 = +2.8% = 0.028

Obstruction = Sta 90+35, Elev = 152.39

Clearance = 20 ft

We use L = 620 ft so obstruction is exactly at EVC.

Thus:

BVC = 87+25 – 3+10 = 84+15

EVC = 87+25 + 3+10 = 90+35

Compute a:

a = [0.028 – (-0.035)] / [2(620)]

a = 0.063 / 1240 = 0.000050806

Elevation at BVC:

z_BVC = 115.78 – (-0.035)(310) – a(310^2)

z_BVC = 115.78 + 10.85 – 4.8829 = 121.7471

Elevation at EVC:

z_EVC = 121.7471 + (-0.035)(620) + a(620^2)

z_EVC = 121.7471 – 21.70 + 19.4775 = 119.5776

Lowest Point

x_LP = -(-0.035) / 2a = 0.035 / 0.000101612

x_LP = 344.5 ft

Station LP = 84+15 + 344.5 = 87+59.5

z_LP = 121.7471 – 0.035(344.5) + a(344.5^2)

z_LP = 121.7471 – 12.0575 + 6.028 = 115.7176

LP = Sta 87+59.5, Elev = 115.718 ft

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