Problem 50 - Algebra 1: Concepts and Skills

Question

Use the following equation which models a cross section of the Barringer Meteor Crater, near Winslow, Arizona. Note that x and y are measured in meters and the center of the crater is at x 0. Barringer Meteor model: \(y=\frac{1}{1800}(x-600)(x+600)\) Assuming the lip of the crater is at y 0, how wide is the crater?

Step-by-Step Solution

Answer

The width of the Barringer Meteor Crater is 1200 meters.

Step 1: Find X-intercepts

To find the X-intercepts (where the graph crosses the x-axis and y=0), set the equation equal to zero and solve for x as follows: \(0=\frac{1}{1800}(x-600)(x+600)\). By setting each factor to zero, the values of x can be found. That is, \(x-600=0\) to get \(x=600\) and \(x+600=0\) to get \(x=-600\).

Step 2: Determine the Width of the Crater

The width of the crater is the difference of the x-coordinates of the points where the graph intersects the x-axis. Mathematically, this can be represented as Width = x₂ - x₁. Here, \(x_1 = -600\) and \(x_2 = 600\). This would be \(Width = 600 - (-600) = 1200\) meters.

Key Concepts

X-intercepts

The concept of X-intercepts is fundamental to understanding quadratic equations and the graphs they generate. An X-intercept is a point where a graph crosses the X-axis, which means at this point, the value of y is zero. When you solve a quadratic equation to find the X-intercepts, essentially, you're finding the points at which the curve touches or crosses the X-axis.

In practice, to find the X-intercepts of the quadratic equation given by the model of the Barringer Meteor Crater, you need to set the entire equation equal to zero and solve for the variable x. This is because the Y-coordinate of an X-intercept is always zero. You'll often end up with two solutions because a quadratic equation can be plotted as a parabola which typically crosses the X-axis at two points. These points provide key information about the shape and position of the parabola.

Quadratic Models

Quadratic models are mathematical tools that use quadratic equations to represent real-world scenarios. They are particularly useful in representing phenomena that can be plotted in parabolic shapes, such as the trajectory of a ball or, as in our exercise, the cross-section of a meteor crater.

The general form of a quadratic equation is \( y = ax^2+bx+c \), where \( a \), \( b \), and \( c \) are constants. For the Barringer Meteor model the equation \( y = \frac{1}{1800}(x - 600)(x + 600) \) is already factored, showing the X-intercepts directly. In real-world terms, quadratic models like this one provide a simple way to visualize and calculate properties such as the width of a meteor crater with clear X-intercepts at \( x = 600 \) and \( x = -600 \) meters.

Mathematical Problem-Solving

Mathematical problem-solving involves understanding the problem, devising a plan, carrying out the plan, and then looking back to review the work done. When presented with a quadratic equation from a real-life scenario, as with the Barringer Meteor Crater, it is not only about finding the solution but also interpreting it correctly within the context.

To solve for the width of the crater, a systematic approach was used. First, the understanding that the lip of the crater is represented by the X-intercepts provided a target for the solution. Then the equation was manipulated algebraically to reveal these X-intercepts. The final step involved subtracting these values to find the total width, demonstrating not just algebraic manipulation but also real-world application of mathematical concepts.