Question

Geography Because of the forces caused by its rotation, Earth is an oblate ellipsoid rather than a sphere. The equatorial radius is 3963 miles and the polar radius is 3950 miles. Find an equation of the ellipsoid. (Assume that the center of Earth is at the origin and that the trace formed by the plane \(z=0\) corresponds to the equator.)

Short answer

The equation of the ellipsoid is \(\frac{x^2}{3963^2} + \frac{y^2}{3963^2} + \frac{z^2}{3950^2} = 1\).

Step-by-step solution

Understanding the formula of an ellipsoid

The general formula for an ellipsoid centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\), where \(x\), \(y\), and \(z\) are the coordinates of a point on the ellipsoid, and \(a\), \(b\), and \(c\) are the radii along the x, y, and z axes, respectively. Since the problem states that the trace formed by the plane \(z=0\) corresponds to the equator, we can identify \(a\) and \(b\) with the equatorial radius and \(c\) with the polar radius.

Plugging in the given values

The problem gives the equatorial radius as 3963 miles and the polar radius as 3950 miles. Therefore, \(a = b = 3963\) and \(c = 3950\). Substituting these values into the ellipsoid equation, we get \(\frac{x^2}{3963^2} + \frac{y^2}{3963^2} + \frac{z^2}{3950^2} = 1\).

Revising the equation

Therefore, the equation of the ellipsoid that represents the Earth, given that the equatorial radius is 3963 miles and the polar radius is 3950 miles, is \(\frac{x^2}{3963^2} + \frac{y^2}{3963^2} + \frac{z^2}{3950^2} = 1\).