# Length of Parabolic Sag Curve with Given Change in Grade Per Station

**Problem**

A grade of -5% is followed by a grade of 1%, the grades intersecting at the vertex (Sta. 10 + 060). The change of grade is restricted to 0.4% in 20 m. Compute the length of the vertical parabolic sag curve in meters.

A. 360 m | C. 300 m |

B. 320 m | D. 340 m |

**Answer Key**

[ C ]

**Solution**

Total change in grade:

$A = g_2 - g_1 = 1 + 5$

$A = g_2 - g_1 = 1 + 5$

$A = 6\%$

Rate of change in grade

$\Delta g = \dfrac{0.4}{20} = 0.02\% ~ \text{per meter}$

Length of Parabolic Sag Curve

$L = \dfrac{A}{\Delta g} = \dfrac{6}{0.02}$

$L = 300 ~ \text{m}$ ← Answer: [ C ]

**On CASIO Calculator: f(x)-991ES PLUS**`MODE → 3:STAT → 2:A+BX`

X | Y |

0 | -5 |

20 | -5 + 0.4 |

`AC`

$L = 1\hat{x} = 300 ~ \text{ft}$ ← Answer: [ C ]

Note:

$\hat{x}$ = `SHIFT → 1:[STAT] → 5:Reg → 4:x-carret`