Let's walk through the solution as shown in your attached image **step-by-step**, explaining each part clearly: --- ### **GIVEN DATA** - **Back Tangent or Initial Grade (G₁):** -2.5% - **Forward Tangent or Final Grade (G₂):** 0.9% - **Length of curve (L):** 425 ft - **Station of VPI:** 28 + 50.00 - **Elevation of VPI (e):** 609.35 ft --- ### **CALCULATIONS** #### **a) Station of BVC (Begin Vertical Curve)** \[ \text{Station of BVC} = \text{VPI Station} - \frac{L}{2 \times 100} \] \[ = 28 + 50 - 2 + 12.50 = 26 + 37.50 \] #### **b) Station of EVC (End Vertical Curve)** \[ \text{Station of EVC} = \text{VPI Station} + \frac{L}{2 \times 100} \] \[ = 28 + 50 + 2 + 12.50 = 30 + 62.50 \] #### **c) Elevation at BVC** \[ \text{Elevation at BVC} = \text{Elevation at VPI} - |G_1| \times \left(\frac{L}{2}\right) \] \[ = 609.35 - 0.025 \times 213 = 614.66 \ \text{ft} \] #### **d) Elevation at EVC** \[ \text{Elevation at EVC} = \text{Elevation at VPI} + G_2 \times \left(\frac{L}{2}\right) \] \[ = 609.35 + 0.009 \times 213 = 611.26 \ \text{ft} \] --- ### **EQUATION OF CURVE** #### **Rate of Slope Change** \[ a = \frac{G_2 - G_1}{2L} \] \[ = \frac{0.009 - (-0.025)}{2 \times 425} = \frac{0.034}{850} = 0.000040000 \] #### **Equation:** \[ y = ax^2 + bx + c \] Where: - \(a = 0.000040000\) - \(b = -0.025\) - \(c = 614.66\) So, \[ y = 0.000040000x^2 - 0.025x + 614.66 \] --- ### **FULL STATIONS** #### **i) Station 27+00, x = 62.5 ft** \[ \text{Elevation} = 0.000040000 \times (62.5)^2 - 0.025 \times 62.5 + 614.66 = \underline{613.25 \ \text{ft}} \] #### **ii) Station 28+00, x = 162.5 ft** \[ \text{Elevation} = 0.000040000 \times (162.5)^2 - 0.025 \times 162.5 + 614.66 = \underline{611.65 \ \text{ft}} \] #### **iii) Station 29+00, x = 262.5 ft** \[ \text{Elevation} = 0.000040000 \times (262.5)^2 - 0.025 \times 262.5 + 614.66 = \underline{610.85 \ \text{ft}} \] #### **iv) Station 30+00, x = 362.5 ft** \[ \text{Elevation} = 0.000040000 \times (362.5)^2 - 0.025 \times 362.5 + 614.66 = \underline{610.85 \ \text{ft}} \] --- ### **h) Location of Lowest/Highest Point** Set derivative to zero: \[ \frac{dy}{dx} = 2ax + b = 0 \] \[ 2 \times 0.000040000 \times x - 0.025 = 0 \] \[ 0.000080000x = 0.025 \implies x = 312.5 \ \text{ft} \] Station: \[ = 26 + 37.50 + 3 + 12.50 = 29 + 50.00 \] --- ### **i) Elevation at Highest/Lowest Point** \[ \text{At } x = 312.5: \] \[ \text{Elevation} = 0.000040000 \times (312.5)^2 - 0.025 \times 312.5 + 614.66 = \underline{610.75 \text{ ft}} \] --- ### **j) Station 29+70, x = 332.5 ft** \[ \text{Elevation} = 0.000040000 \times (332.5)^2 - 0.025 \times 332.5 + 614.66 = \underline{610.77 \ \text{ft}} \] --- ### **k) From BVC, x = 250 ft** \[ \text{Elevation} = 0.000040000 \times (250)^2 - 0.025 \times 250 + 614.66 = \underline{610.91 \ \text{ft}} \] --- ## **Summary of Key Answers** - **Equation of curve:** \( y = 0.000040000 x^2 - 0.025x + 614.66 \) - **Station of BVC:** \( 26 + 37.50 \) - **Station of EVC:** \( 30 + 62.50 \) - **Elevation at BVC:** \( 614.66 \) ft - **Elevation at EVC:** \( 611.26 \) ft - **Elevation at full stations (27+00, 28+00, 29+00, 30+00):** 613.25 ft, 611.65 ft, 610.85 ft, 610.85 ft - **Station of lowest/highest point:** 29+50.00 - **Elevation at lowest/highest point:** 610.75 ft --- If you need a specific calculation clarified or a step explained in more detail, let me know!

Let's analyze the problem step-by-step with clear calculations and explanations. --- ### **Given Data:** - Length of vertical curve, \( L = 425.00\, \text{ft} \) - Grade before VPI, \( g_1 = -2.30\% = -0.023 \) - Grade after VPI, \( g_2 = +0.90\% = +0.009 \) - VPI station, \( \text{STA}_{VPI} = 23 + 50.00 = 23.50\, \text{stations} \) - VPI elevation, \( E_{VPI} = 949.35\, \text{ft} \) --- ### **Step 1: Determine the station of BVC and EVC** The BVC (Begin Vertical Curve) is located at: \[ \text{STA}_{BVC} = \text{VPI station} - \frac{L}{2} \times \frac{1}{100} = 23.50 - \frac{425}{2} \times 0.01 \] \[ = 23.50 - 212.5 \times 0.01 = 23.50 - 2.125 = \boxed{21.375} \] The EVC (End Vertical Curve) is at: \[ \text{STA}_{EVC} = 23.50 + 2.125 = \boxed{25.625} \] --- ### **Step 2: Elevations at BVC and EVC** - Elevation at BVC: \[ E_{BVC} = E_{VPI} + g_1 \times \frac{L}{2} \] \[ = 949.35 + (-0.023) \times 212.5 = 949.35 - 4.8875 = \boxed{944.4625\, \text{ft}} \] - Elevation at EVC: \[ E_{EVC} = E_{VPI} + g_2 \times \frac{L}{2} \] \[ = 949.35 + 0.009 \times 212.5 = 949.35 + 1.9125 = \boxed{951.2625\, \text{ft}} \] --- ### **Step 3: Equation of the curve** The vertical curve elevation \( y \) at a distance \( x \) from BVC (where \( x=0 \) at BVC) is: \[ y = a x^2 + b x + c \] with: - \( a = \frac{g_2 - g_1}{2L} \) - \( b = g_1 \) - \( c = E_{BVC} \) Calculate \( a \): \[ a = \frac{0.009 - (-0.023)}{2 \times 425} = \frac{0.032}{850} \approx 3.7647 \times 10^{-5} \] Set \( c = 944.4625 \) ft (elevation at BVC). --- ### **Step 4: Elevations on the curve at full stations** - Full stations are measured relative to BVC (x=0 at BVC): | Station | \( x \) (ft) | Calculation | Elevation \( y \) (ft) | |------------|--------------|------------------------------|---------------------------| | 22+00 | \( -1 \times 100 = -100 \) | \( y = a(-100)^2 + b(-100) + c \) | \( y = 3.7647 \times 10^{-5} \times 10,000 - 0.023 \times 100 + 944.4625 \) | \( 3.7647 \times 10^{-5} \times 10,000 = 0.3765 \) | \( -2.3 + 944.4625 + 0.3765 = 942.5385 \) | | 23+50 | \( +50 \) | \( y = a(50)^2 + b(50) + c \) | \( 3.7647 \times 10^{-5} \times 2,500 + 0.009 \times 50 + 944.4625 \) | \( 0.0941 + 0.45 + 944.4625 = 944.9966 \) | | 25+00 | \( +150 \) | \( y = a(150)^2 + b(150) + c \) | \( 3.7647 \times 10^{-5} \times 22,500 + 0.009 \times 150 + 944.4625 \) | \( 0.849 + 1.35 + 944.4625 = 946.6615 \) | | 25+50 | \( +200 \) | \( y = a(200)^2 + b(200) + c \) | \( 3.7647 \times 10^{-5} \times 40,000 + 0.009 \times 200 + 944.4625 \) | \( 1.505 + 1.8 + 944.4625 = 947.7675 \) | --- ### **Step 5: Elevations on the tangents at full stations** - Tangent at BVC (grade \( g_1 = -0.023 \)): \[ E_{tangent} = E_{BVC} + g_1 \times (x - 0) \] - At full stations: | Station | \( x \) (ft) | Elevation \( E \) | Calculation | Result | |------------|--------------|--------------------------|--------------|---------| | 22+00 | -100 | \( 944.4625 - 0.023 \times 100 \) | \( 944.4625 - 2.3 = 942.1625 \) | | 23+50 | 50 | \( 944.4625 + 0.009 \times 50 \) | \( 944.4625 + 0.45 = 944.9125 \) | | 25+00 | 150 | \( 944.4625 + 0.009 \times 150 \) | \( 944.4625 + 1.35 = 945.8125 \) | | 25+50 | 200 | \( 944.4625 + 0.009 \times 200 \) | \( 944.4625 + 1.8 = 946.2625 \) | - Tangent at EVC (grade \( g_2= +0.009 \)): \[ E_{tangent} = E_{EVC} + g_2 \times (x - 150) \] for points beyond EVC. --- ### **Step 6: Tangent offsets at full stations** Offset = \( y_{curve} - y_{tangent} \) | Station | \( x \) | \( y_{curve} \) | \( y_{tangent} \) | Offset | |------------|--------------|---------------------|---------------------|---------| | 22+00 | -100 | 942.5385 | 942.1625 | 0.376 | | 23+50 | 50 | 944.9966 | 944.9125 | 0.0841 | | 25+00 | 150 | 946.6615 | 945.8125 | 0.849 | | 25+50 | 200 | 947.7675 | 946.2625 | 1.505 | --- ### **Step 7: High/Low point station and elevation** - The high/low point occurs where the first derivative \( dy/dx = 0 \): \[ dy/dx = 2a x + b = 0 \] \[ x = - \frac{b}{2a} = - \frac{g_1}{2a} = - \frac{-0.023}{2 \times 3.7647 \times 10^{-5}} \approx \frac{0.023}{7.5294 \times 10^{-5}} \approx 305.4\, \text{ft} \] - The station of the highest/lowest point: \[ \text{Station} = \text{VPI station} + x = 23.50 + \frac{x}{100} \approx 23.50 + 3.054 = \boxed{26.554} \] - Elevation at this point: \[ y = a x^2 + b x + c \] \[ = 3.7647 \times 10^{-5} \times (305.4)^2 - 0.023 \times 3.054 + 944.4625 \] Calculate: \[ x^2 \approx 93354 \] \[ a x^2 \approx 3.7647 \times 10^{-5} \times 93354 \approx 3.517 \] \[ b x \approx -0.023 \times 3.054 \approx -0.070 \] \[ \text{Elevation} \approx 944.4625 + 3.517 - 0.070 = \boxed{948.0095\, \text{ft}} \] --- ### **Step 8: Elevation at station 29+70** - \( x \) from BVC: \[ X = 29.70 - 21.375 = 8.325\, \text{stations} \Rightarrow x = 8.325 \times 100 = 832.5\, \text{ft} \] - Elevation: \[ y = a x^2 + b x + c \] \[ = 3.7647 \times 10^{-5} \times (832.5)^2 - 0.023 \times 8.325 + 944.4625 \] Calculate: \[ x^2 \approx 693,056 \] \[ a x^2 \approx 3.7647 \times 10^{-5} \times 693,056 \approx 26.095 \] \[ b x \approx -0.023 \times 8.325 \approx -0.191 \] \[ \text{Elevation} \approx 944.4625 + 26.095 - 0.191 \approx \boxed{970.366\, \text{ft}} \] --- ### **Step 9: Elevation at 250 ft from BVC** - \( x = 250\, \text{ft} \) - Elevation: \[ y = a(250)^2 + b(250) + c \] \[ = 3.7647 \times 10^{-5} \times 62,500 - 0.023 \times 250 + 944.4625 \] Calculate: \[ a x^2 = 3.7647 \times 10^{-5} \times 62,500 = 2.353 \] \[ b x = -0.023 \times 250 = -5.75 \] \[ \text{Elevation} = 944.4625 + 2.353 - 5.75 = \boxed{941.0655\, \text{ft}} \] --- ## **Final Answers:** | **Item** | **Result** | |------------|--------------| | **e. Elevations at full stations:** | 22+00 ≈ 942.54 ft, 23+50 ≈ 944.99 ft, 25+00 ≈ 946.66 ft, 25+50 ≈ 947.77 ft | | **f. Elevations on tangents:** | 22+00 ≈ 942.16 ft, 23+50 ≈ 944.91 ft, 25+00 ≈ 945.81 ft, 25+50 ≈ 946.26 ft | | **g. Tangent offsets:** | ~0.38 ft, 0.084 ft, 0.85 ft, 1.51 ft | | **h. High/low point station:** | approx. **26+33.54** (or station 26+34) | | **i. High/low point elevation:** | approx. **948.01 ft** | | **j. Elevation at station 29+70:** | approx. **970.37 ft** | | **k. Elevation 250 ft from BVC:** | approx. **941.07 ft** | --- If you need further clarification or specific calculations, feel free to ask!

Certainly! Let's carefully revisit the first question based on the provided image solution and proceed step-by-step with detailed explanations. --- ### **Given Data:** - Length of the vertical curve, \( L = 425\, \text{ft} \) - Grade before VPI, \( g_1 = -2.30\% = -0.023 \) - Grade after VPI, \( g_2 = +0.90\% = +0.009 \) - VPI station, \( \text{STA}_{VPI} = 23 + 50.00 = 23.50\, \text{stations} \) - VPI elevation, \( E_{VPI} = 949.35\, \text{ft} \) --- ### **Step 1: Find station of BVC and EVC** - **BVC (Begin Vertical Curve):** \[ \text{Station of BVC} = \text{VPI station} - \frac{L}{2} \times \frac{1}{100} = 23.50 - \frac{425}{2} \times 0.01 \] \[ = 23.50 - 212.5 \times 0.01 = 23.50 - 2.125 = \boxed{21.375\, \text{stations}} \] - **EVC (End Vertical Curve):** \[ \text{Station of EVC} = 23.50 + 2.125 = \boxed{25.625\, \text{stations}} \] --- ### **Step 2: Calculate elevations at BVC and EVC** - **Elevation at BVC:** \[ E_{BVC} = E_{VPI} + g_1 \times \frac{L}{2} = 949.35 + (-0.023) \times 212.5 \] \[ = 949.35 - 4.8875 = \boxed{944.4625\, \text{ft}} \] - **Elevation at EVC:** \[ E_{EVC} = E_{VPI} + g_2 \times \frac{L}{2} = 949.35 + 0.009 \times 212.5 \] \[ = 949.35 + 1.9125 = \boxed{951.2625\, \text{ft}} \] --- ### **Step 3: Equation of the curve** The shape of the parabola: \[ y = a x^2 + b x + c \] where: - \( a = \frac{g_2 - g_1}{2L} = \frac{0.009 - (-0.023)}{2 \times 425} = \frac{0.032}{850} \approx 3.7647 \times 10^{-5} \) - \( b = g_1 = -0.023 \) - \( c = E_{BVC} = 944.4625\, \text{ft} \) --- ### **Step 4: Calculate elevations at full stations** Full stations are measured relative to the BVC station (x=0 at BVC): | Station | \( x \) (ft) | Calculation | Elevation \( y \) (ft) | |---------|--------------|------------------------------|----------------------------| | 22+00 | \( -100 \) | \( y = a(-100)^2 + b(-100) + c \) | \( 3.7647 \times 10^{-5} \times 10,000 - 0.023 \times 100 + 944.4625 \) | | 23+50 | \( +50 \) | \( y = a(50)^2 + b(50) + c \) | \( 3.7647 \times 10^{-5} \times 2,500 - 0.023 \times 50 + 944.4625 \) | | 25+00 | \( +150 \) | \( y = a(150)^2 + b(150) + c \) | \( 3.7647 \times 10^{-5} \times 22,500 - 0.023 \times 150 + 944.4625 \) | | 25+50 | \( +200 \) | \( y = a(200)^2 + b(200) + c \) | \( 3.7647 \times 10^{-5} \times 40,000 - 0.023 \times 200 + 944.4625 \) | Calculations: - For 22+00 (\( x = -100 \)): \[ y = 0.000037647 \times 10,000 - 2.3 + 944.4625 = 0.3765 - 2.3 + 944.4625 = 942.5385\, \text{ft} \] - For 23+50 (\( x = 50 \)): \[ y = 0.000037647 \times 2,500 - 1.15 + 944.4625 = 0.0941 - 1.15 + 944.4625 = 943.4066\, \text{ft} \] - For 25+00 (\( x = 150 \)): \[ y = 0.000037647 \times 22,500 - 3.45 + 944.4625 = 0.849 - 3.45 + 944.4625 = 942.8615\, \text{ft} \] - For 25+50 (\( x = 200 \)): \[ y = 0.000037647 \times 40,000 - 4.6 + 944.4625 = 1.505 - 4.6 + 944.4625 = 941.3675\, \text{ft} \] --- ### **Step 5: Elevations on the tangents at full stations** - **Tangent at BVC (grade \( g_1 = -0.023 \)):** \[ E_{tangent} = E_{BVC} + g_1 \times x \] | Station | \( x \) (ft) | Elevation \( E_{tan} \) (ft) | |---------|--------------|------------------------------| | 22+00 | -100 | \( 944.4625 - 2.3 = 942.1625 \) | | 23+50 | 50 | \( 944.4625 + 0.009 \times 50 = 944.4625 + 0.45 = 944.9125 \) | | 25+00 | 150 | \( 944.4625 + 0.009 \times 150 = 944.4625 + 1.35 = 945.8125 \) | | 25+50 | 200 | \( 944.4625 + 0.009 \times 200 = 944.4625 + 1.8 = 946.2625 \) | --- ### **Summary:** | **Full Station** | **Elevation on Curve (ft)** | **Elevation on Tangent (ft)** | **Tangent Offset (ft)** | |------------------|------------------------------|------------------------------|-------------------------| | 22+00 | 942.54 | 942.16 | 0.38 | | 23+50 | 943.41 | 944.91 | -1.50 | | 25+00 | 942.86 | 945.81 | -2.95 | | 25+50 | 941.37 | 946.26 | -4.89 | *(Note: Offsets are curve elevation minus tangent elevation)* --- ### **Step 6: Highest/Lowest Point (Vertex of parabola)** - **Position:** \[ x_{vertex} = - \frac{b}{2a} = - \frac{-0.023}{2 \times 3.7647 \times 10^{-5}} \approx \frac{0.023}{7.5294 \times 10^{-5}} \approx 305.4\, \text{ft} \] - **Station:** \[ \text{Station} = \text{VPI station} + \frac{x_{vertex}}{100} = 23.50 + 3.054 = \boxed{26+33.54} \] - **Elevation at vertex:** \[ y_{max/min} = a x^2 + b x + c \] Calculations: \[ x^2 = (305.4)^2 \approx 93354 \] \[ a x^2 = 3.7647 \times 10^{-5} \times 93354 \approx 3.517 \] \[ b x = -0.023 \times 3.054 \approx -0.070 \] \[ \text{Elevation} \approx 944.4625 + 3.517 - 0.070 \approx \boxed{948.01\, \text{ft}} \] --- ### **Final notes:** - **Equation of the curve:** \[ \boxed{ y = 3.7647 \times 10^{-5} x^2 - 0.023 x + 944.4625 } \] - **Station of BVC:** 21.375 - **Station of EVC:** 25.625 - **Elevations at key points are as calculated above.** --- **This completes the detailed re-evaluation of the solution based on the image-provided data.** If you need any further clarification, feel free to ask!