Question

Use the following equation, are of the Barringer Meteor Crater, near Winslow, Arizona. D Source: Jet Propulsion Laboratory Barringer Meteor Crater model: \(y=\frac{1}{1800}(x-600)(x+600)\) Assuming the lip of the crater is at \(y=0,\) how wide is the crater?

Short answer

The width of the Barringer Meteor Crater is 1200 units.

Step-by-step solution

Identify the equation

We are given the equation for the Barringer Meteor Crater model as \(y=\frac{1}{1800}(x-600)(x+600)\). We need to find the width of the crater, which means we need to find the x-values in this equation when y=0.

Solve the equation for x when y=0

Let's insert y=0 into the equation and solve for x. We get 0=\(\frac{1}{1800}(x-600)(x+600)\). To solve this equation we simply set each factor to zero because a product is zero when at least one of the factors is zero. This gives us two equations: x-600=0 and x+600=0.

Find the x-values

Solving the above equations gives us x=600 and x=-600.

Calculate the width of the crater

We know that these x values represent two points along the x axis, the width of the crater is therefore the distance between these two points which is found by subtracting one x value from the other. So we have width = \(x_{1} - x_{2} = 600 - (-600) = 1200\) . Therefore, the crater is 1200 units wide .

Key concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is a fundamental skill in algebra. When dealing with quadratics, these equations can often represent real-world scenarios, like the size of a meteor crater or the trajectory of a launched rocket. A quadratic equation typically has the form \( ax^2 + bx + c = 0 \) where \( a \) , \( b \) , and \( c \) are coefficients, and \( x \) represents the unknown variable. The solutions to these equations are known as the roots or zeros of the equation.

To solve a quadratic equation, it's crucial to isolate \( x \) and determine its value(s). There are several methods to do this, including factoring, completing the square, using the quadratic formula, or graphing. For instance, in the example of the Barringer Meteor Crater model, \( y=\frac{1}{1800}(x-600)(x+600) \) was solved by setting \( y \) to zero and finding the \( x \) values that satisfy the equation. This is because the reality represented by \( y = 0 \) usually pertains to a specific condition, like the lip of a crater being at ground level.

Factoring Quadratics

Factoring quadratics is a critical technique when solving quadratic equations, especially when they are set to zero. A factored form of a quadratic equation is where you express it as the product of its binomials or monomial factors, like \( (x-p)(x-q)=0 \) where \( p \) and \( q \) are the roots. The beauty of factoring lies in its simplicity since once in factored form, the zero product property can be applied directly.

Consider the Barringer Meteor Crater equation \( y=\frac{1}{1800}(x-600)(x+600) \). It is already in factored form, showing the roots at \( x=600 \) and \( x=-600 \) easily. To factor quadratics not yet in factored form, one must look for two numbers that multiply to give the constant term (the product of \( a \) and \( c \) in \( ax^2 + bx + c \) ) and add to give the coefficient of the linear term ( \( b \) in the same equation). Factoring is often the first resort in solving quadratics because it's usually quicker and doesn't require complex calculations as other methods do.

Zero Product Property

The zero product property is an essential rule in algebra that states if a product of factors equals zero, then at least one of the factors must also be zero. This property is applied after factoring a quadratic equation and is fundamental in finding the solution(s) to the equation.

In practice, once you factor the quadratic equation to something like \( (x-p)(x-q) = 0 \), you can then set each factor equal to zero: \( x-p = 0 \) or \( x-q = 0 \)). You then solve these simple equations to find the values of \( x \) that make the original equation true. From the Barringer Meteor Crater problem, after setting the equation \( \frac{1}{1800}(x-600)(x+600) = 0 \) with \( y \) as zero, we applied the zero product property to find the crater's width. The roots \( x=600 \) and \( x=-600 \) are found by setting each factor of the product equal to zero. The zero product property is an excellent example of how algebraic concepts help us draw conclusions about physical phenomena.