TIME The expression \( \pi(r-3)^{2} \) represents the area covered by the hour hand on a clock in one rotation, where \( r \) is the radius of the entire clock. Write a polynomial that represents the area covered by the hour hand in one rotation.

To write a polynomial that represents the area covered by the hour hand in one rotation, let's start by expanding the given expression: \[ \pi(r - 3)^2 \] **Step 1: Expand the Binomial** \[ (r - 3)^2 = r^2 - 6r + 9 \] **Step 2: Multiply by π** \[ \pi(r^2 - 6r + 9) = \pi r^2 - 6\pi r + 9\pi \] **Final Polynomial** \[ \pi r^2 - 6\pi r + 9\pi \] **Interpretation:** - **\(\pi r^2\)**: Represents the area of the entire clock. - **\(-6\pi r\)**: Adjusts for the length of the hour hand. - **\(+9\pi\)**: Adds a constant area based on the adjustment. So, the polynomial that represents the area covered by the hour hand in one rotation is: \[ \boxed{\pi r^{2} - 6\,\pi\, r + 9\,\pi} \]