A road’s vertical profile is modeled by the elevation function
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A road’s vertical profile is modeled by the elevation function E(x)=200+50tan(0.02x), where E(x) is measured in meters above sea level and x is the horizontal distance, in meters, from the starting point of the road (with 0≤x<2π/0.02 so the expression is defined).
(a) Determine an explicit formula for E′(x).
(b) Compute E′(500) and interpret its value in the context of the road’s construction (include appropriate units).
(c) The maximum safe design slope for this type of road is 0.12 meters of elevation per meter of horizontal distance. For what interval of x in meters does the road’s slope remain within the allowable limit? Present your answer as an inequality and justify your reasoning.