Problem 70 - College Algebra Essentials

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed a nonlinear system that modeled the orbits of Earth and Mars, and the graphs indicated the system had a solution with a real ordered pair.

Step-by-Step Solution

Answer

The statement does not make sense because in terms of the physical context being modeled (the orbits of Earth and Mars), these paths do not intersect. Thus, it's not plausible for the system to have a solution represented by a real ordered pair that signifies intersection.

Step 1: Understanding the Problem

The problem involves a system of nonlinear equations that model the orbits of Earth and Mars. In this case, a solution represents a point where the orbits of Earth and Mars intersect. A real ordered pair (x,y) refers to a particular point in the Cartesian coordinate system (often representing time and distance, but it can stand for any two variables).

Step 2: Reasoning

Given that planets follow definite paths around the sun, such paths can indeed be modelled by functions which when graphed will give certain trajectories. If the orbits did intersect at a particular point, we would have a solution represented by a real ordered pair. This just represents a particular place and time where both planets are at the same place. However, in reality, the orbits of Earth and Mars do not intersect, hence it wouldn't make sense to say that such a system has a solution in terms of the physical situation being modelled.

Step 3: Conclusion

So while mathematically it is possible for a system of nonlinear equations to have a solution in terms of a real ordered pair, in the context of these equations modeling the orbits of the Earth and Mars, it's not physically plausible because those paths do not intersect.

Key Concepts

System of Equations

Understanding a system of equations is crucial in solving mathematical problems, especially when modeling real-world situations. A system of equations consists of multiple equations with multiple variables. When you solve this system, you're looking for a set of values for the variables that satisfy all the equations simultaneously.

• These equations can be linear or nonlinear.
• Linear systems often have straightforward solutions, typically lines on a graph that intersect at a common point.
• Nonlinear systems, however, involve more complex curves such as circles or ellipses.

For instance, in the case of modeling planetary orbits, we use nonlinear equations. Here, you're not just looking for where simple lines meet but rather where curves like ellipses intersect. Solutions in this context are real ordered pairs that represent points at which these curves intersect. Keep in mind, though, that in our universe, the physical interpretation of these solutions must make sense.

Orbits of Planets

When studying the motion of planets, we often refer to their orbits, which are the paths they travel around the sun. These paths are indeed real-life examples of nonlinear curves.

• The orbits are elliptical, a type of curve studied in nonlinear equations.
• Each planet’s orbit can be analyzed using mathematical functions that can predict positions over time.
• Although these orbits are theoretically intersection points in math, physically, planets don’t actually collide.

Mars and Earth, for instance, have their unique orbital paths, and hence they never physically intersect. Mathematically, you could determine an intersection point if you examine a system of equations modeling the orbits, but this point won't have a real physical counterpart because orbits do not cross one another in reality.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane that provides a way to describe locations.

• It uses two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).
• Each point on the plane is identified by a pair of numbers \(x, y\).
• This system is fundamental in plotting graphs and solving equations, including those modeling complex phenomena like planetary orbits.

When you solve a system of equations in the Cartesian coordinate system, you're finding where curves intersect, essentially looking for common solutions expressed as ordered pairs. This not only helps in theoretical analysis but also in understanding real-world phenomena via graphical representations. The simplicity of representing points, lines, and curves makes it an indispensable part of mathematics, especially in elucidating complex systems like planetary orbits.