Math Problem Statement

If the orbit of the moon can be modeled using the equation \( \frac{x^2}{63,500} + \frac{y^2}{50,900} = 1 \), what is the shape of the moon's orbit?

Solution

The equation given is:

x263,500+y250,900=1\frac{x^2}{63,500} + \frac{y^2}{50,900} = 1

This resembles the standard equation for an ellipse:

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where aa and bb are the semi-major and semi-minor axes of the ellipse, respectively. In this case, since the values under x2x^2 and y2y^2 are different (63,50063,500 and 50,90050,900), it indicates that the orbit is elliptical.

Therefore, the correct answer is Elliptical.

Would you like more details or explanations? Here are 5 related questions you might find interesting:

  1. What are the differences between a circular and elliptical orbit?
  2. How does the eccentricity of an ellipse affect the shape of an orbit?
  3. What is the significance of the semi-major and semi-minor axes in an elliptical orbit?
  4. How do Kepler's laws relate to elliptical orbits?
  5. What other celestial bodies follow elliptical orbits?

Tip: In an elliptical orbit, the distance between the orbiting body and the focal points (like the Earth or Sun) changes, which is why planets and moons speed up and slow down at different points in their orbit.

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Math Problem Analysis

Mathematical Concepts

Conic Sections
Ellipses
Geometry

Formulas

Equation of an ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Theorems

Ellipse: A conic section with two foci, where the sum of distances from any point on the ellipse to the foci is constant.

Suitable Grade Level

Grades 10-12