- Analyze the clock as a circle divided into 12 equal sections of \(30^\circ\) each.
- At 4:00, the hands form a central angle of \(4 \times 30^\circ = 120^\circ\).
- Apply the sector area formula \(A = \frac{\theta}{360^\circ} \pi r^2\) with \(r=9\) and \(\theta=120^\circ\).
- Simplify the expression to find the final area: \(\boxed{27 \pi \text{ in}^2}\).
Explanation
- Analyze the clock face and determine the central angle.
The clock face is a circle with a radius \(r = 9\) inches. We need to find the area of the sector formed by the hour hand and the minute hand at 4:00.
A full circle represents \(360^\circ\). A clock is divided into 12 equal segments by the hour marks, meaning each segment covers \(360^\circ / 12 = 30^\circ\).
At 4:00, the minute hand points directly at the 12, and the hour hand points directly at the 4. The number of segments between the 12 and the 4 is 4.
Thus, the central angle \(\theta\) between the hands is \(4 \times 30^\circ = 120^\circ\).
- Calculate the area of the sector using the formula.
The formula for the area of a sector of a circle with radius \(r\) and central angle \(\theta\) (in degrees) is given by:
\[A = \frac{\theta}{360^\circ} \pi r^2\]
Substituting the known values, where \(\theta = 120^\circ\) and \(r = 9\) inches:
\[A = \frac{120^\circ}{360^\circ} \pi (9)^2\]
First, simplify the fraction \(\frac{120}{360}\):
\[\frac{120}{360} = \frac{1}{3}\]
Next, calculate the square of the radius:
\[9^2 = 81\]
Now, compute the area:
\[A = \frac{1}{3} \pi (81) = 27 \pi \text{ in}^2\]
- Final Conclusion
The area of the sector formed by the hands of the clock at 4:00 is \(27 \pi \text{ in}^2\). Comparing this to the provided options, it matches the third option.
The final answer is \(\boxed{27 \pi \text{ in}^2}\).
Examples
Understanding sector areas is essential for various circular designs, such as creating a pie chart to visualize data or calculating the area of a specific slice of a circular garden or pizza. For instance, if you have a circular patio and want to install a fountain that covers a specific wedge-shaped portion, you would use this sector area formula to determine the exact amount of materials needed based on the radius and the central angle of that wedge. This application ensures efficiency and precision in design and construction projects.