an ant is walking around an ellipse. the ant starts at the top and walks around in a clockwise direction and…

Explanation:

Step1: Find the center of the ellipse

The center of the ellipse can be found by averaging the (x -)coordinates and (y -)coordinates of the vertices. The vertices of the ellipse seem to be at key points. The center of the ellipse in the (x -)direction is (x_0 = 1) (by observing the symmetry of the ellipse about (x = 1)). In the (y -)direction, the center (y_0=\frac{9+( - 1)}{2}=4).

Step2: Determine the general form of parametric equations for an ellipse

The general form of parametric equations for an ellipse centered at ((x_0,y_0)) with semi - major axis (a) and semi - minor axis (b) and a full - rotation in (T) minutes (where the parameter is (t) in minutes) is (x(t)=x_0 + a\cos(\omega t)) and (y(t)=y_0 + b\sin(\omega t)), where (\omega=\frac{2\pi}{T}). Here (T = 2\pi), so (\omega = 1).

Step3: Find the semi - axes

The distance from the center ((1,4)) to the left - most point ((-2,4)) gives the semi - minor axis in the (x -)direction. The distance (a=\vert1-( - 2)\vert=3). The distance from the center ((1,4)) to the top - most point ((1,9)) gives the semi - major axis in the (y -)direction. The distance (b=\vert9 - 4\vert = 5). Since the ant starts at the top, for the (x) - coordinate, we use (x(t)=1+3\cos(t)) (the cosine function starts at its maximum value when (t = 0) for the (x) - displacement from the center in this case). For the (y) - coordinate, we use (y(t)=4 + 5\sin(t)) (the sine function starts at (0) when (t = 0) and we want the starting (y) - value to be the (y) - coordinate of the top - most point relative to the center).

Answer:

(x(t)=1 + 3\cos(t)), (y(t)=4+5\sin(t))