Question
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.
Ask by Willis Stuart. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The area covered by the hour hand in one rotation is represented by the polynomial:
\[
\pi r^{2} - 6\pi r + 9\pi
\]
Solution
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:
\[
\boxed{\pi r^{2} - 6\,\pi\, r + 9\,\pi}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the area covered by the hour hand, we start with the expression \( \pi(r-3)^{2} \). To expand this, we can use the formula for squaring a binomial: \( (a-b)^{2} = a^{2} - 2ab + b^{2} \). Here, let \( a = r \) and \( b = 3 \): \[ (r-3)^{2} = r^{2} - 2 \cdot r \cdot 3 + 3^{2} = r^{2} - 6r + 9. \] Now, we multiply this expansion by \( \pi \): \[ \pi(r-3)^{2} = \pi(r^{2} - 6r + 9) = \pi r^{2} - 6\pi r + 9\pi. \] Thus, the polynomial that represents the area covered by the hour hand in one rotation is: \[ \pi r^{2} - 6\pi r + 9\pi. \]
