≤ft{egin{array}{l}x=6(t-sin t), y=6(1-cos t),end{array} y=9,(0<x<12 π, y ≥ 9) ight.

The given equations represent a parametric curve defined by the equations \( x = 6(t - \sin t) \) and \( y = 6...

1. : - The equation for \( x \) suggests that as \( t \) increases, \( x \) will also increase, but it is influenced by the term \( -6\sin t \), which oscillates between -6 and 6. This means that \( x \) will have a periodic behavior influenced by the sine function. - The equation for \( y \) indicates that \( y \) is always non-negative and oscillates between 0 and 12, since \( \cos t \) oscillates between -1 and 1. Specifically, \( y \) reaches its minimum value of 0 when \( \cos t = 1 \) (i.e., \( t = 0, 2\pi, 4\pi, \ldots \)) and its maximum value of 12 when \( \cos t = -1 \) (i.e., \( t = \pi, 3\pi, 5\pi, \ldots \)). 2. : - The curve will oscillate vertically between \( y = 0 \) and \( y = 12 \) as \( t \) varies. - The condition \( y \geq 9 \) implies that we are only interested in the portions of the curve where \( y \) is at least 9. This occurs when \( 1 - \cos t \geq 1.5 \), or equivalently when \( \cos t \leq -0.5 \). This corresponds to the intervals where \( t \) is in the ranges \( \frac{2\pi}{3} + 2k\pi \) to \( \frac{4\pi}{3} + 2k\pi \) for integer \( k \). 3. : - The condition \( 0 x 12\pi \) limits the values of \( t \). Since \( x = 6(t - \sin t) \), we can analyze the behavior of \( t - \sin t \) to find the corresponding range of \( t \) that satisfies this condition. - The curve will have segments where it oscillates above the line \( y = 9 \) and will be periodic due to the nature of the sine and cosine functions. - The maximum value of \( y \) (12) occurs at specific intervals of \( t \), while the minimum value of \( y \) (0) is not of interest due to the condition \( y \geq 9 \). - The values of \( x \) will be constrained by the oscillation of \( t - \sin t \), which will affect how many complete cycles fit within the range \( 0 x 12\pi \). The parametric equations describe a periodic curve that oscillates vertically, with specific segments above the line \( y = 9 \). The horizontal extent of the curve is limited to \( 0 x 12\pi \), which corresponds to specific intervals of the parameter \( t \). Please feel free to ask any questions related to this description or analysis!