$\"\"$

The parametric equations are $\"\"$ and $\"\"$ .

The tip of the hour-hand will complete one full rotation around the clock between $\"\"$ clock noon and $\"\"$ midnight.

It will complete another full rotation around the clock between $\"\"$ midnight and $\"\"$ noon the next day.

Therefore, the hour-hand will complete $\"\"$ full rotations around the clock from $\"\"$ noon to $\"\"$ noon the next day.

Since the parametric equations are written in terms of the trigonometric functions $\"\"$ and $\"\"$ one full rotation will be completed every $\"\"$ , which is the period of these two functions.

Therefore, $\"\"$ full rotations will be completed after $\"\"$ .

So, an interval for $\"\"$ in radians that can be used to describe the motion of the tip is $\"\"$ .

$\"\"$

(1) Draw the coordinate plane.

(2) Graph the parametric equations $\"\"$ and $\"\"$ .

Graph :

$\"\"$ .

$\"\"$

The parametric equations are $\"\"$ and $\"\"$ .

$\"\"$

From the trignometric identity : $\"\"$ .

$\"\"$

A rectangular equation that models the motion of the hour-hand is $\"\"$ .

This equation is in standard form, so $\"\"$ .

Thus, the radius of the circle traced out by the hour-hand is $\"\"$ .

$\"\"$

An interval for $\"\"$ in radians that can be used to describe the motion of the tip is $\"\"$ .

Graph :

$\"\"$

A rectangular equation that models the motion of the hour-hand is $\"\"$ .