Calculus - Applications of Differentiation question?

Question

A section of highway connecting two hillsides with grades of 6% and 4% is to be build between two points that are separated by a horizontal distance of 2000 feet. At the point where the two hillsides come together, there is a 50-foot difference in elevation.

a) Design a section of highway connecting the hillsides modeled by the function f(x) = ax^3 + bx^2 + cx + d (-1000 less than or equal to x less than or equal to 1000). At the points A and B, the slope of the model must match the grade of the hillside.
b) Use a graphing utility to graph the model.
c) Use a graphing utility to graph the derivative of the model.
d) Determine the grade at the steepest part of the transitional section of the highway.

I need to show work step-by-step for this, so please format your answer as such. Thanks! :)

Answer 1

The "grade" is the slope: rise/run.

Let A be the point on the hill with a grade of 6%, while B is the point on the hill with a grade of 4%. Also let A correspond to x = -1000, while B corresponds to x = 1000. (Note that x corresponds to distance from the midpoint of the section of highway)

The problem doesn't tell you whether A is higher than B or vice versa, so I assume you can choose whichever you wish, as long as you make your choice explicit.

Your function has 4 unknowns (a, b, c, and d) so we need 4 equations.

If A is the higher point, then two equations are:
f(-1000) = 50
f(1000) = 0

otherwise, reverse the values.

Since the slope must match the grades of the hillsides:
f'(-1000) = 0.06
f'(1000) = 0.04

So you have 4 linear equations in 4 unknowns and so can solve for a, b, c, and d. (Note these equations are linear because the unknowns appear as degree 1. The fact that x appears squared and cubed doesn't matter because x is known.)

Parts b and c depend on your graphing utility and you probably don't use mine.

For part d, you are looking for the maximum value f'(x) over the range [-1000, 1000]. It will occur either at the endpoints or at a point where f''(x) = 0.

So compute f''(x), which will be a linear function. Then compute the x for which f''(x) is 0. Then compute f'(x) at that point and compare it with the slope at the ends.