Problem 47 - Algebra 2

Question

The supporting cables of the Golden Gate Bridge approximate the shape of a parabola. The parabola can be modeled by \(y=0.00012 x^{2}+6,\) where \(x\) represents the distance from the axis of symmetry and \(y\) represents the height of the cables. The related quadratic equation is \(0.00012 x^{2}+6=0\). What does the discriminant tell you about the supporting cables of the Golden Gate Bridge?

Step-by-Step Solution

Answer

The discriminant is negative, indicating no real solutions; the cables never meet the horizontal plane (x-axis).

Step 1: Identify the coefficients

The given quadratic equation is \(0.00012 x^{2} + 6 = 0\). First, we need to express it in the standard form \(ax^2 + bx + c = 0\). Here, \(a = 0.00012\), \(b = 0\), and \(c = 6\).

Step 2: Formula for the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\).

Step 3: Substitute the values into the Discriminant formula

Using the values \(a = 0.00012\), \(b = 0\), and \(c = 6\), substitute them into the discriminant formula: \(D = 0^2 - 4(0.00012)(6)\).

Step 4: Calculate the Discriminant

Calculate \(D = 0 - 4(0.00012)(6)\). This simplifies to \(D = -0.00288\).

Step 5: Interpret the Discriminant

Since the discriminant \(D = -0.00288\) is negative, this indicates that the quadratic equation has no real solutions. This implies that the parabola does not intersect the x-axis.

Key Concepts

The Discriminant and Its Importance

In quadratic equations like the one modeling the Golden Gate Bridge cables, the discriminant plays a crucial role. It helps us determine the nature of the roots of the equation. The formula to find the discriminant in the equation \( ax^2 + bx + c = 0 \) is \( D = b^2 - 4ac \).

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution, or a repeated real solution.
• If \( D < 0 \), there are no real solutions, as the roots are complex.

In our Golden Gate Bridge model, the given equation is \( 0.00012 x^2 + 6 = 0 \). Here, \( a = 0.00012 \), \( b = 0 \), and \( c = 6 \). Substituting these values into the discriminant formula yields \( D = 0 - 4(0.00012)(6) = -0.00288 \). As this result is negative, it tells us that the equation has no real solutions. This is confirmed by the fact that the parabola does not touch or cross the x-axis, symbolized by the real plane.

Understanding Parabolic Modeling

Parabolic modeling is used in many fields, such as physics, engineering, and architecture, just like in the design of the Golden Gate Bridge. A parabola is a U-shaped curve that can model the path of an object or shape of structures. In our specific scenario, the equation \( y = 0.00012x^2 + 6 \) represents the height of the cables above the ground, with \( x \) marking the horizontal distance from the axis of symmetry of the bridge. In these cases:

• The coefficient \( a = 0.00012 \) tells us about the "curvature" or "width" of the parabola. A smaller \( a \) value means a wider parabola.
• The positive sign indicates the parabola opens upwards.
• The constant \( c = 6 \) represents the y-intercept where the parabola meets the y-axis.

This model helps engineers and architects to understand and predict the structure's behavior under different conditions, ensuring safety and integrity.

Exploring Real Solutions in Quadratic Equations

Real solutions, also known as roots, of a quadratic equation \( ax^2 + bx + c = 0 \), correspond to the points where the graphed parabola intersects the x-axis. Finding these real solutions can aid in multiple analyses, such as trajectory predictions, motion equations, and structural assessments.

• If you get a positive discriminant \( D > 0 \), two real solutions manifest, implying intersection at two points.
• With a discriminant equal to zero \( D = 0 \), the parabola just touches the x-axis, indicating a singular real solution.
• A negative discriminant \( D < 0 \) means the parabola doesn't intersect the x-axis, signifying no real solutions.

In our exercise concerning the Golden Gate Bridge, the discriminant was negative, signaling no real x-intercepts. Hence, the supporting cables neatly arch over any axis intersections, creating a stable, visually stunning design.