Let's go step by step for your vertical curve problem. ### **Given Data** - **L** = 425.00 ft (Length of vertical curve) - **g₁** = -2.30% (Initial tangent grade) - **g₂** = +0.90% (Final tangent grade) - **VPI Sta** = 23+50.00 ft (Station of Vertical Point of Intersection) - **VPI Elev** = 949.35 ft (Elevation at VPI) Let’s define: - **BVC** (Beginning of Vertical Curve) - **EVC** (End of Vertical Curve) --- ## **Step 1: Find BVC and EVC Stations** - The BVC is **L/2** before the VPI: - \( \text{BVC Sta} = 23+50.00 - \frac{425}{2} = 23+50.00 - 212.50 = 21+37.50 \) - The EVC is **L/2** after the VPI: - \( \text{EVC Sta} = 23+50.00 + 212.50 = 25+62.50 \) --- ## **Step 2: Find Elevation at BVC** - Distance from BVC to VPI: 212.50 ft - Grade at BVC: g₁ = -2.30% = -0.0230 \[ \text{Elev at BVC} = \text{Elev at VPI} - (\text{distance}) \times g_1 \] \[ = 949.35\,\text{ft} - 212.5 \times (-0.0230) \] \[ = 949.35 + 4.8875 = 954.2375\,\text{ft} \] --- ## **Step 3: Vertical Curve Equation** The parabolic vertical curve equation (for station x feet past BVC): \[ y = y_{\text{BVC}} + g_1 x + \frac{A}{2L} x^2 \] where - \( A = g_2 - g_1 = 0.90\% - (-2.30\%) = 3.20\% = 0.0320 \) - \( L = 425 \) - \( y_{\text{BVC}} = 954.2375 \) \[ y = 954.2375 + (-0.0230)x + \frac{0.0320}{2 \times 425} x^2 \] \[ \frac{0.0320}{850} = 0.00003765 \] So, \[ y = 954.2375 - 0.0230x + 0.00003765x^2 \] --- ## **Step 4: Full Stations from BVC to EVC** BVC = 21+37.50 EVC = 25+62.50 Full stations are: 22+00, 23+00, 24+00, 25+00 ### **For each full station, find:** - Distance from BVC, \( x \) - Elevation on curve - Elevation on tangent - Tangent offset --- ### **22+00** - Sta: 22+00 = 2200 ft - \( x_{22+00} = 2200 - 2137.5 = 62.5 \) ft ### **23+00** - Sta: 23+00 = 2300 ft - \( x_{23+00} = 2300 - 2137.5 = 162.5 \) ft ### **24+00** - Sta: 24+00 = 2400 ft - \( x_{24+00} = 2400 - 2137.5 = 262.5 \) ft ### **25+00** - Sta: 25+00 = 2500 ft - \( x_{25+00} = 2500 - 2137.5 = 362.5 \) ft --- ## **Step 5: Calculations** ### **e. Elevations on the curve at full stations** \[ y = 954.2375 - 0.0230x + 0.00003765x^2 \] #### **22+00 (x = 62.5):** \[ y = 954.2375 - 0.0230 \times 62.5 + 0.00003765 \times (62.5)^2 \\ = 954.2375 - 1.4375 + 0.147 \\ = 952.9470\,\text{ft} \] #### **23+00 (x = 162.5):** \[ y = 954.2375 - 0.0230 \times 162.5 + 0.00003765 \times (162.5)^2 \\ = 954.2375 - 3.7375 + 0.992 \\ = 951.4920\,\text{ft} \] #### **24+00 (x = 262.5):** \[ y = 954.2375 - 0.0230 \times 262.5 + 0.00003765 \times (262.5)^2 \\ = 954.2375 - 6.0375 + 2.591 \\ = 950.7910\,\text{ft} \] #### **25+00 (x = 362.5):** \[ y = 954.2375 - 0.0230 \times 362.5 + 0.00003765 \times (362.5)^2 \\ = 954.2375 - 8.3375 + 4.944 \\ = 950.8440\,\text{ft} \] --- ### **f. Elevations on the tangents at full stations** (The tangent in question is the initial tangent, unless specified otherwise.) \[ y_{\text{tangent}} = 954.2375 + g_1 x \] #### **22+00 (x = 62.5):** \[ = 954.2375 + (-0.0230 \times 62.5) = 954.2375 - 1.4375 = 952.8000\,\text{ft} \] #### **23+00 (x = 162.5):** \[ = 954.2375 - 3.7375 = 950.5000\,\text{ft} \] #### **24+00 (x = 262.5):** \[ = 954.2375 - 6.0375 = 948.2000\,\text{ft} \] #### **25+00 (x = 362.5):** \[ = 954.2375 - 8.3375 = 945.9000\,\text{ft} \] --- ### **g. Tangent offsets at full stations** \[ \text{Offset} = \text{Elevation (curve)} - \text{Elevation (tangent)} \] #### **22+00:** \[ 952.9470 - 952.8000 = 0.1470\,\text{ft} \] #### **23+00:** \[ 951.4920 - 950.5000 = 0.9920\,\text{ft} \] #### **24+00:** \[ 950.7910 - 948.2000 = 2.5910\,\text{ft} \] #### **25+00:** \[ 950.8440 - 945.9000 = 4.9440\,\text{ft} \] --- ### **h. High/Low Point Station** The high/low point occurs where the slope is zero: \[ g_1 + \frac{A}{L} x = 0 \implies x = -\frac{g_1 L}{A} \] \[ x = -\frac{-0.0230 \times 425}{0.0320} = \frac{9.775}{0.0320} = 305.47\,\text{ft} \] \[ \text{Station} = \text{BVC Sta} + x = 21+37.50 + 305.47 = 21+37.50 + 3+05.47 = 24+42.97 \] --- ### **i. High/Low Point Elevation** Plug \( x = 305.47 \) into the curve equation: \[ y = 954.2375 - 0.0230 \times 305.47 + 0.00003765 \times (305.47)^2 \] \[ = 954.2375 - 7.0258 + 3.501 = 950.7127\,\text{ft} \] --- ### **j. Elevation at station 29+70.00** Station 29+70.00 = 2970 ft \[ x = 2970 - 2137.5 = 832.5\,\text{ft} \] But EVC is at 2562.5, so x = 832.5 is **beyond the EVC** (the curve only covers up to x = 425 ft). So at this station, the elevation is on the final tangent. Final tangent equation (from EVC): - Final tangent grade: \( g_2 = +0.0090 \) - Elev at EVC \( y_{\text{EVC}} \): \[ x_{\text{EVC}} = 425.0\,\text{ft} \] \[ y_{\text{EVC}} = 954.2375 - 0.0230 \times 425 + 0.00003765 \times 425^2 \] \[ = 954.2375 - 9.775 + 6.803 \\ = 951.2655\,\text{ft} \] Distance from EVC to 29+70.00: \[ 2970 - 2562.5 = 407.5\,\text{ft} \] \[ y = y_{\text{EVC}} + g_2 \times (\text{distance from EVC}) \] \[ = 951.2655 + 0.0090 \times 407.5 = 951.2655 + 3.6675 = 954.9330\,\text{ft} \] --- ### **k. Elevation on the curve at a point 250.00 ft from the BVC** \( x = 250.00 \) ft \[ y = 954.2375 - 0.0230 \times 250.00 + 0.00003765 \times (250.00)^2 \] \[ = 954.2375 - 5.75 + 2.353 = 950.8405\,\text{ft} \] --- # **Summary Table** | Sta | x (ft) | Elev (Curve) | Elev (Tangent) | Offset | |--------|--------|--------------|----------------|---------| |22+00 | 62.5 | 952.95 | 952.80 | 0.147 | |23+00 | 162.5 | 951.49 | 950.50 | 0.99 | |24+00 | 262.5 | 950.79 | 948.20 | 2.59 | |25+00 | 362.5 | 950.84 | 945.90 | 4.94 | --- ## **Final Answers** ### **e. Elevations on the curve at full stations** - 22+00: **952.95 ft** - 23+00: **951.49 ft** - 24+00: **950.79 ft** - 25+00: **950.84 ft** ### **f. Elevations on the tangents at full stations** - 22+00: **952.80 ft** - 23+00: **950.50 ft** - 24+00: **948.20 ft** - 25+00: **945.90 ft** ### **g. Tangent offsets at full stations** - 22+00: **0.15 ft** - 23+00: **0.99 ft** - 24+00: **2.59 ft** - 25+00: **4.94 ft** ### **h. High/low point station** - **24+42.97** ### **i. High/low point elevation** - **950.71 ft** ### **j. Elevation at station 29+70.00** - **954.93 ft** (on the final tangent) ### **k. Elevation 250 ft from BVC** - **950.84 ft** --- **Let me know if you need the calculations for other stations or explanations for any particular step!**

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