Calculus Bridge Application Question
A section of highway connecting two hillsides with grades of 6% and 4% is to be built between two points that are separated by a horizontal distance of 2000 feet. At the point where two hillsides come together there is a 50-foot difference in elevation. Design a section of highway connecting the hillsides modeled by the cubic function f(x)=ax^3+bx^2+cx+d (-1000 is less than or equal to x which is less than or equal to 1000). At the points A and B the slope of the model must match the grade of the hillside. Find a derivative function for the model. Determine the grade at the steepest part of the transitional section of the highway.