Problem 46 - 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 \(0.00012 x^{2}+6=0\). Calculate the value of the discriminant.

Step-by-Step Solution

Answer

The discriminant is -0.00288.

Step 1: Identify components of the quadratic equation

The quadratic equation given is \[ 0.00012x^{2} + 6 = 0. \]In the standard form \(ax^{2} + bx + c = 0\), we identify \(a = 0.00012\), \(b = 0\), and \(c = 6\).

Step 2: Understand the 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 values into the discriminant formula

Plug in the values from the identified components into the discriminant formula:\[ D = (0)^{2} - 4 \cdot 0.00012 \cdot 6. \]

Step 4: Calculate the value of the discriminant

Calculate the expression:\[ D = 0 - 4 \times 0.00012 \times 6. \] First, multiply \(0.00012\) by \(6\) to get \(0.00072\). Then, multiply this by \(4\) to get \(0.00288\). Therefore, the discriminant is \[ D = -0.00288. \]

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically written in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). It represents a fundamental concept in algebra, describing the curve known as a parabola. Quadratic equations can have 0, 1, or 2 real roots, determined by the discriminant. This discriminant, calculated as \(D = b^2 - 4ac\), informs us about the nature of the roots:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is one real root (also called a double root).
• If \(D < 0\), there are no real roots but two complex roots.

Understanding and calculating the discriminant is key to solving quadratic equations and predicting the properties of their solutions.

Parabola

A parabola is a U-shaped curve that graphically represents a quadratic equation. It plays a critical role in various real-world applications like satellite dishes and bridge cables. The general algebric formula for a parabola is \(y = ax^2 + bx + c\). Parabolas can open upwards or downwards depending on the sign of the coefficient \(a\):

• If \(a > 0\), the parabola opens upward.
• If \(a < 0\), the parabola opens downward.

In the context of the Golden Gate Bridge, the cables form a parabolic shape that helps to equally distribute weight and minimize structural stress. Understanding this curve helps engineers design stable and efficient structures.

Golden Gate Bridge

The Golden Gate Bridge is an iconic suspension bridge that spans the San Francisco Bay, using parabolically shaped cables to support its massive structure. Each cable forms a parabola, providing both aesthetic appeal and functional strength. This shape ensures weight distribution is even across the bridge, preventing localized stress that could lead to damage. The quadratic model \(y = 0.00012x^2 + 6\) describes the parabola of the cables, linking mathematics to real-world engineering solutions. This understanding illustrates the vital connection between algebraic concepts and engineering designs.

Axis of Symmetry

In a parabola, the axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It can be determined from the quadratic equation \(ax^2 + bx + c = 0\) using the formula for the vertex, \(x = -\frac{b}{2a}\). This calculation highlights a key property of parabolas: symmetry. In the equation with \(b = 0\), such as the one modeling the Golden Gate Bridge, the axis of symmetry is simply \(x = 0\). Understanding this line helps in analyzing and graphing parabolas, ensuring accurate representations of their shapes. The axis of symmetry is critical to studying motion and forces in structures like the Golden Gate Bridge.