#### Problem (1)

PVI station and location are 74+26.50 and 772.30 ft. What are the endpoints' stations and elevations of a 750.00 ft long curve with incoming and outgoing grades of -4.00% and +2.00%? #### Problem (2)

PVI station and location are 56+50.00 and 1325.00 ft. Incoming and outgoing grades are +5.0% and -3.5%. A 500.00 ft long vertical curve will be used.  ##### (a) Curve Table
 Station di, sta Elevi, ft gi, % 59+00 EVC 5.0000 1316.250 -3.5 58+00 4.0000 1318.900 -1.8 57+00 3.0000 1319.850 -0.1 56+00 2.0000 1319.100 +1.6 55+00 1.0000 1316.650 +3.3 54+00 BVC 0.0000 1312.500 +5.0

Math checks in red.

##### (b) Highest and lowest points

Because this is a crest curve, the lowest point will occur at either the BVC or EVC.
In this case lowest point is at BVC: 54+00.00 @ 1312.50 ft

Highest: #### Problem (3)

A 425.00 ft long curve ends at station and elevation 88+00.00 and 1220.00. Its initial and final grades are -4.00% and +2.75%.

##### (a) Curve table
 Station di, sta Elevi, ft gi, % 92+25 EVC 4.2500 1217.344 +2.75 92+00 4.0000 1216.706 +2.35 91+00 3.0000 1216.147 +0.76 90+00 2.0000 1215.176 -0.82 89+00 1.0000 1216.794 -2.41 88+00 BVC 0.0000 1220.000 -4.00

Math checks in red.

##### (b) Highest and lowest points

Sag curve so either BVC or EVC is highest point
Highest: BVC, 88+00.00 @ 1220.000 ft

Lowest: 90+51.852 @ 1214.963 ft

#### Problem (4)

A curve begins at station 32+00.00 and elevation 912.00 ft; it ends at station 38+00.00 and elevation 915.00 ft. The PVI elevation is 918.00 ft.

##### (a) Curve table
 Station di, sta Elevi, ft gi, % 38+00 EVC 6.0000 915.000 -1.00 37+00 5.0000 915.750 -0.50 36+00 4.0000 916.000 0.00 35+00 3.0000 915.750 +0.50 34+00 2.0000 915.000 +1.00 33+00 1.0000 913.750 +1.50 32+00 BVC 0.0000 912.000 +2.00

Math checks in red.

##### (b) Highest and lowest points

This is crest curve so the lowest point will be at BVC or EVC.
Lowest point: BVC, 32+00 @ 912.000 ft
Highest point occurs at grade = 0.00%: 36+00 @ 916.000 ft

#### Problem (5)

A 400.00 ft vertical curve begins at station 29+00.00 and elevation 954.00 ft. Its initial and final grades are -1.00% and -2.25%.

#### Part (a)

What are the station and elevation of the curve's midpoint?

The midpoint is not the PVI. It has the same station as the PVI, but is located on the curve.  Alternate solution #### Part (b)

What are the stations and elevations of the curve's highest and lowest points?

Because both grades are negative, the highest point is the BVC, lowest is EVC.
Highest: 29+00.00 @ 954.00
Lowest: Hits: 7215