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 (see figure). 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 \(f(x)=a x^{3}+b x^{2}+c x+d\) \((-1000 \leq x \leq 1000)\). 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.

Short answer

The solution involves setting up a system of equations using the given conditions and the equation of the cubic function, then solving the system to find the coefficients. The function and its derivative can be graphed using a graphing utility, and the steepest part of the transitional section is found by finding the maximum of the derivative within the given range.

Step-by-step solution

Analyze the Problem

An expression for \(f(x)\) will be found which satisfies the conditions provided in the problem. We are given two hillsides separated by a horizontal distance of 2000 feet and a vertical difference of 50 feet. Our function \(f(x)\) must calculate the elevation of the road at any point between the hillsides, with the specific requirements that the slope matches the hillsides' grades at points A and B, and \(f(-1000)=0\), \(f(1000)=50\). Grades of 6% and 4% are equivalent to slopes of 0.06 and 0.04 respectively.

Setup the System of Equations

To find the function \(f(x)\), the coefficients \(a\), \(b\), \(c\), and \(d\) must be determined using the information provided. Create a system of equations using the formulas for the slope and the points \(A\) and \(B\). We know the slope at \(A\) is \(0.06\), the slope at \(B\) is \(0.04\), \(f(-1000)=0\), and \(f(1000)=50\). So we get four equations: \[f(-1000)= a(-1000)^3 + b(-1000)^2 + c(-1000) + d = 0\] \[f(1000) = a(1000)^3 + b(1000)^2 + c(1000) + d = 50\] \[f'(-1000) = 3a(-1000)^2 + 2b(-1000) + c = 0.06\] \[f'(1000) = 3a(1000)^2 + 2b(1000) + c = 0.04\]

Solve the System of Equations

This system of equations can be solved either manually or by using a software such as MATLAB or Python. The solution will give the values of \(a\), \(b\), \(c\), and \(d\).

Graph the Function

By substituting the values of \(a\), \(b\), \(c\), and \(d\), we get the actual function \(f(x)\). To graph this function, a graphing tool like GeoGebra or a graphing calculator can be used.

Graph the Derivative

The derivative of the function can be graphed using the same method, but replacing \(f(x)\) with \(f'(x) = 3ax^2 + 2bx + c\).

Determine the Maximum Grade

To find the steepest grade of the transitional section of the highway, we find the maximum value of \(f'(x)\) in the interval \([-1000, 1000]\). This will represent the maximum slope of the road and therefore, the steepest grade, which can be found by differentiating \(f'(x)\) and solving for \(x\) within the given range.

Key concepts

Calculus in Engineering

Calculus is an indispensable tool in engineering, providing methods for modeling and solving complex problems that involve change and motion. For instance, in civil engineering, the design of structures such as highways must account for various factors, including terrain slopes, which can be effectively tackled using calculus. In a practical scenario, calculus helps to define the shape of a highway that must span uneven ground with varying slopes.

When an engineer is tasked with constructing a road between two points with differing elevations and grades, calculus allows for the precise calculation of the necessary curvature of the road to create a smooth transition. This involves using integrals for finding areas under curves that represent slopes (grades) and derivatives to measure the rate at which these slopes change over horizontal distances. The language of calculus can accurately capture these complex relationships within mathematical functions that will then dictate the structures to be built in the real world.

System of Equations

In our exercise, we dealt with a system of equations. This is a collection of two or more equations with a set of unknowns that we attempt to solve simultaneously. In our highway grade problem, we utilized a system of equations to determine the coefficients of a cubic polynomial. This polynomial function represents the elevation at any given point along the highway's path.

The system of equations was constructed from known conditions, such as specific function values at certain points (the elevations at the endpoints), and derivatives at these points (the given slopes or grades). When the coefficients of the polynomial are found, the function describing the road's curve can be modeled. Engineers and scientists use systems of equations frequently to characterize multiple relationships between variables in real-world scenarios.

Derivative Graphing

Graphing the derivative of a function is crucial in understanding the behavior of the function itself, especially when it comes to rates of change. In the context of the highway problem, the gradient at every point along the road was needed. The derivative graph tells us how the slope varies as we move along the highway. The steeper the slope of this graph, the steeper is the grade of the road at that point.

The exercise called for the graphing of the derivative of the highway's elevation model. This graphical analysis helps in identifying crucial elements such as maximum and minimum slopes, giving a visual representation that can guide engineers in making design decisions to meet safety and practicality standards. Being able to graph derivatives is a valuable skill for predicting and addressing potential issues before the physical construction begins.

Maximum Value of a Function

Finding the maximum value of a function within a specific interval is a common task in optimization problems, and it's exactly what was needed to determine the steepest grade of the transitional section of the highway. This involves finding the largest output value of the function, which represents the slope in our case.

In the final step of our problem, derivatives played a central role again. By analyzing the derivative of the highway's slope function, we can determine where the slope is at its greatest within the interval. This is critical for engineers who need to assess the maximum grade to ensure that the design adheres to safety regulations. The process of finding the maximum value includes testing endpoints and locating critical points where the derivative equals zero or is undefined, then comparing the values of the function at these points.