A \(100-\mathrm{m}\)-long track is to be built \(50 \mathrm{~m}\) wide. If it is to be elliptical, what equation could describe it if the 100 -m length is along the \(x\)-axis? a) \(50 x^{2}+100 y^{2}=1000\) c) \(4 x^{2}+y^{2}=2500\) b) \(2 x^{2}+y^{2}=250\) d) \(x^{2}+2 y^{2}=250\)

To comprehend the elliptical shape, it's essential to identify its semi-major and semi-minor axes, which are akin to the radius in a circle, but for an ellipse, which is stretched. The semi-major axis, denoted by 'a', represents the longest radius of the ellipse, generally aligning with the x-axis when the ellipse is in its standard position. The semi-minor axis, represented by 'b', indicates the shortest radius and aligns with the y-axis. An ellipse's semi-major and semi-minor axes are crucial as they define the overall shape and size of the ellipse.

When working with a problem involving an ellipse, like designing a track, identifying these axes is the first step. It is critical to understand that the semi-major axis is half the total length of the ellipse, and the semi-minor axis is half the width. This understanding is foundational in deriving the equation to describe an elliptical shape, as it tells us how far the curve stretches in each direction from the center.

The general equation of an ellipse with its center at the origin (0,0) is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \.\] Here, '\(x\text{ and }y\)' are the coordinates of any point on the ellipse, while '\(a\)' and '\(b\)' are the lengths of the semi-major and semi-minor axes, respectively. This equation is derived from the fact that an ellipse is the set of all points for which the sum of the distances to two fixed points (the foci) is constant.

To fully grasp the concept behind this mathematical beauty, one can visualize stretching a circle in one direction. The circle's equation transforms accordingly, with '\(a^2\)' and '\(b^2\)' in the denominators balancing the stretch. When plotted, points that satisfy this equation form the classic, oval shape of an ellipse. Understanding this equation is pivotal for solving problems related to ellipses, as it allows one to determine the precise location of points along the curved shape and dictates the boundary of the elliptical space.

In the context of elliptical track design, the equation and dimensions of an ellipse become quite practical. For example, in the provided exercise, a track is to be designed with an elliptical shape, 100 meters long and 50 meters wide. We utilize our knowledge of semi-major and semi-minor axes to begin plotting this track. Given the lengths, we can determine that the semi-major axis 'a' is 50 meters and the semi-minor axis 'b' is 25 meters.

Applying these dimensions to the general equation of the ellipse results in a tailored equation for the track: \[ \frac{x^2}{50^2} + \frac{y^2}{25^2} = 1 \.\] Multiplying through by the least common multiple of '\(a^2\)' and '\(b^2\)', in this case, 2500, simplifies the equation, leading to an equation that suits the design needs. This process highlights the importance of a systematic approach to designing elliptical structures, ensuring the final track accurately reflects the specified dimensions and serves as a perfect elliptical running path for athletes.