Which of the following is a mathematically defined surface that approximates the shape of the Earth?

The ellipsoid is the correct choice because it is a mathematically defined surface that accurately represents the Earth's shape. The Earth is not a perfect sphere due to its rotation, which causes it to bulge slightly at the equator. As a result, the ellipsoidal model incorporates these variations and provides a more precise approximation of the Earth's geometry.

The ellipsoid takes into account the flattening at the poles and the bulging at the equator, which is critical for applications in geodesy, navigation, and mapping. This model allows for more accurate calculations involving distance, area, and volume on the Earth's surface.

In contrast, while a sphere may give a basic representation, it oversimplifies the Earth's actual shape. A cylinder does not correspond to the Earth’s geometry at all, and a cube is even less representative of the Earth’s surface, as it introduces unnatural angles and flat surfaces that are not found in nature. The ellipsoid model is therefore the most suitable and scientifically accepted representation of the Earth's shape.