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Magazine Chapter 14: Problem 37
Set up systems of equations and solve them graphically. Assume earth is a sphere, with \(x^{2}+y^{2}=41\) as the equation of a circumference (distance in thousands of \(\mathrm{km}\) ). If a meteorite approaching earth has a path described as \(y^{2}=20 x+140,\) will the meteorite strike earth? If so, where?
Short Answer
Expert verified
No, the meteorite will not strike Earth; the equations do not intersect at valid points.
Step by step solution
01
Identify Equations
We have two equations to consider for this problem. The first one is the equation of Earth, represented as the circumference: \(x^2 + y^2 = 41\). The second one is the path of the meteorite, described as a parabola: \(y^2 = 20x + 140\).
02
Setting the System of Equations
To find the intersection points, which represent potential collision points, we need to solve these two equations simultaneously:\[ \begin{cases} x^2 + y^2 = 41 \ y^2 = 20x + 140 \end{cases}\] We will substitute the expression for \( y^2 \) from the second equation into the first equation.
03
Substitute and Simplify
Using \(y^2 = 20x + 140\), substitute this into \(x^2 + y^2 = 41\). This gives: \(x^2 + (20x + 140) = 41\). Simplifying this gives: \(x^2 + 20x + 140 = 41\). Further simplification leads to: \(x^2 + 20x + 99 = 0\).
04
Solve the Quadratic Equation
Now, we solve the quadratic equation \(x^2 + 20x + 99 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 20\), and \(c = 99\), we calculate: \(x = \frac{-20 \pm \sqrt{20^2 - 4 \times 1 \times 99}}{2}\). Simplifying, we find: \(x = \frac{-20 \pm \sqrt{400 - 396}}{2}\) which simplifies to: \(x = \frac{-20 \pm \sqrt{4}}{2}\). Therefore, \(x = -10 \pm 1\). The solutions are \(x = -9\) and \(x = -11\).
05
Find Corresponding y Values
Substitute \(x = -9\) and \(x = -11\) back into the equation \(y^2 = 20x + 140\). For \(x = -9\), \(y^2 = 20(-9) + 140 = -180 + 140 = -40\), which is not possible since \(y^2\) cannot be negative. For \(x = -11\), \(y^2 = 20(-11) + 140 = -220 + 140 = -80\), which is also not possible. These results suggest a miscalculation or no intersection physically valid on Earth.
06
Interpreting Results
Since substituting \(x = -9\) and \(x = -11\) resulted in invalid answers for \(y^2\) (as \(y^2\) cannot be negative), this suggests that the path of the meteorite does not intersect Earth's surface at these coordinates, physically or graphically. Thus, graphically checking these outcomes validates no collision.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are characterized by the presence of an \(x^2\) term, which gives the graph of a quadratic equation a distinct parabolic shape. Quadratic equations can be solved using various methods, such as factoring, completing the square, or applying the quadratic formula:
- The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Inside the formula, the term \(b^2 - 4ac\) is called the discriminant, which indicates the nature of the roots.
- If \(b^2 - 4ac > 0\), there are two distinct real roots. If \(b^2 - 4ac = 0\), there is exactly one real root. If \(b^2 - 4ac < 0\), the equation has no real roots, only complex ones.
Graphical Solution
Graphical solutions involve solving equations by using graphs to visually identify the points of intersection. This method is particularly helpful for understanding the relationships between two different equations, such as a linear equation and a quadratic equation, by plotting their graphs on a coordinate plane.
- The intersection points indicate the solutions to the equations when solved simultaneously.
- Graphical solutions provide an intuitive understanding of how different mathematical relationships can interact.
- When graphing, ensure accurate plots by selecting appropriate scales for your axes. This makes it easier to identify intersection points.
Simultaneous Equations
Simultaneous equations are sets of equations with multiple variables that are solved together to find a common solution for the variables involved. When dealing with simultaneous equations, the solution is a point where all the equations in the system are satisfied. In the context of our problem:
- We are solving \(x^2 + y^2 = 41\) and \(y^2 = 20x + 140\) to determine if a meteorite will hit the Earth.
- The goal is to find values of \(x\) and \(y\) that satisfy both equations simultaneously.
- Common methods include substitution, elimination, and graphical analysis.
Parabola
A parabola is a U-shaped graph that represents a quadratic function or relation. In our exercise, the path of the meteorite is described by a quadratic equation, which takes the shape of a parabola. Some important attributes of a parabola include:
- The vertex, which is the highest or lowest point on the graph, depending on the orientation of the parabola.
- The axis of symmetry, a vertical line that runs through the vertex, dividing the parabola into two symmetrical halves.
- The focus and directrix, geometric properties that further describe the parabola's shape and position.
