# Calculus

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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! :)

• Calculus -

You need to find the four constants, a, b, c, and d.

you have a value for f(x) at x = -1000
write it out:
f(-1000) = -1000 = a (-1000)^3 b(-1000)^2 + c (-1000) + d

you have a value for f(x) at x = +1000
write it out:
f(+1000) = a (1000)^3 + b (1000)^2 + c (1000) + d

you have a value for the slope, f'(x)
at x = -1000
f'(-1000) = 3 a (-1000)^2 + 2 b (-1000) + c

you have a value for the slope at x =+1000
f'(1000) = 3 a (1000)^2 + 2 b (1000) + c

That is four linear equations with four unknowns, a, b, c, and d

Solve those four equations simultaneously and you will have the function and its derivative which you can graph.

The point where the absolute value of the derivative is maximum is the steepest part.

You can find that extreme by setting the derivative of the derivative equal to zero.
6 a x + 2 b = 0
solve for x and calculate the slope at that point.

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