Problem 54 - Algebra for College Students

Question

Solve each problem. A parabolic arch 27 feet high spans a parkway. How wide is the arch if the center section of the parkway, a section that is 50 feet wide, has a minimum clearance of 15 feet? 75 feet

Step-by-Step Solution

Answer

The total width of the parabolic arch is 75 feet.

Step 1: Understand the Problem

We need to determine the total width of the parabolic arch, given that the height of the arch is 27 feet and a section in the middle of the parkway spanning 50 feet has a minimum clearance of 15 feet.

Step 2: Set Up the Coordinate System

Place the vertex of the parabola at the origin \(0,0\) for simplicity. The equation of the parabola will have the form \(y = ax^2 + 27\). The endpoints of the 50-foot section are at \(x = -25\) and \(x = 25\). At these points, the height of the parabola is 15 feet.

Step 3: Establish the Equation for Minimum Clearance

At the points \(-25, 15\)\ and \(25, 15\), substitute into the parabola equation: a(-25)^2 + 27 = 15\ and a(25)^2 + 27 = 15\. Solve for \(a\): 625a + 27 = 15, .which simplifies to a = -0.0192\.

Step 4: Determine the Width of the Arch

The parabola intersects the x-axis when \(y = 0\). Solve \(ax^2 + 27 = 0\) for \(x\) using \(a = -0.0192\): -0.0192x^2 + 27 = 0\ implies x^2 = \frac{27}{0.0192}\. After computations, the solution for \(x\) is approximately \(-37.5, 37.5\) feet.

Key Concepts

Coordinate System

When tackling a problem involving a parabolic arch, it's crucial to set up an appropriate coordinate system. This helps you visualize and solve the equations effectively. In our scenario, we place the vertex of the parabola at the origin, which is the point \(0,0\). This simplifies many calculations since the parabola's highest point is at the origin.
By placing our coordinates wisely, we make our algebraic work much easier. The equation of our parabola then becomes dependent on the vertical stretch, which is controlled by the coefficient \(a\) in the equation \(y = ax^2 + 27\). This setup means every other calculation revolves around these key coordinates and helps in understanding the problem's spatial constraints more intuitively.
Using this coordinate system, you can easily spot where the parabola intersects the y-axis, x-axis, or any other lines, critical in determining attributes of the parabolic path.

Parabolic Arch

A parabolic arch forms the shape of our problem's pathway. It is a symmetric curve represented by a quadratic equation. In our case, the equation \(y = ax^2 + 27\) describes this mirror-like shape arching over the parkway.
The vertex, here located at \(0,27\), is the peak of the arch. The symmetry of a parabolic arch is helpful because once you know one side, you automatically know the other side. Each foot of width on one side is mirrored on the other. This is why quirks and properties of the parabola, like its vertex, can tell us so much about its behavior and trajectory.
The given problem specifies certain points on this parabolic path, specifically a section that is 50 feet wide with a minimum height of 15 feet. These coordinates help in establishing the true form of this arch by determining the value of \(a\) in the quadratic equation.

Minimum Clearance

Minimum clearance in the context of a parabolic arch refers to the smallest vertical space available under the arch. This is particularly important for infrastructure, where vehicles or pedestrians might pass through.
Here, the problem specifies a minimum clearance of 15 feet across a 50-foot section, which lies centrally under the arch spanning between \(x = -25\) and \(x = 25\). This information is pivotal in establishing the parabola's equation, particularly the coefficient \(a\), which adjusts the arch's curvature to ensure it meets this clearance requirement.
Through substituting these clearance points into the parabolic equation, we determine the exact shape of the curve. The minimum clearance confirms the parabola's limits, allowing us to establish where it meets the necessary height across its span and ensuring safe passage underneath.