Question

what is the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00?
○ 6.75π in.²
○ 20.25π in.²
○ 27π in.²
○ 81π in.²

Explanation:

Step1: Find the central angle at 4:00

A clock is a circle (360°), with 12 hours. So each hour represents $\frac{360^{\circ}}{12} = 30^{\circ}$ per hour. At 4:00, the hour and minute hands are 4 hours apart. So the central angle $\theta = 4\times30^{\circ}=120^{\circ}$. Convert to radians: $\theta=\frac{120^{\circ}}{180^{\circ}}\pi=\frac{2}{3}\pi$ radians. Or use the fraction of the circle: $\frac{4}{12}=\frac{1}{3}$ of the circle? Wait, no: 4 hours out of 12, so the fraction is $\frac{4}{12}=\frac{1}{3}$? Wait, no, 360° total, 4 hours: 430=120°, which is $\frac{120}{360}=\frac{1}{3}$? Wait, no, 120/360 is 1/3? Wait, 1203=360, yes. Wait, but let's check the sector area formula: $A = \frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta$ is the central angle in degrees, or $A=\frac{1}{2}r^{2}\theta$ in radians.

Step2: Calculate the sector area

Using the degree formula: $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$. Here, $\theta = 4\times30^{\circ}=120^{\circ}$, $r = 9$ inches. So $A=\frac{120^{\circ}}{360^{\circ}}\times\pi\times(9)^{2}=\frac{1}{3}\times\pi\times81 = 27\pi$? Wait, no, wait: 120/360 is 1/3? Wait, 120 divided by 360 is 1/3? Yes. Then 1/3 of 81π is 27π? But wait, let's check the answer options. Wait, maybe I made a mistake. Wait, at 4:00, the hour hand is at 4, minute at 12. The number of hours between them is 4, so the angle is 4(360/12)=120 degrees. The sector area formula is (θ/360)πr². So θ=120, r=9. So (120/360)π81 = (1/3)81π = 27π? But wait, the options include 27π. Wait, but let's re - check. Wait, maybe I messed up the fraction. Wait, 4 hours out of 12, so the fraction is 4/12 = 1/3. So sector area is 1/3 of the circle's area. The circle's area is πr²=π81=81π. Then 1/3 of 81π is 27π. So that's one of the options. Wait, but let's check the other way. Wait, maybe the angle is 120 degrees, and using the formula A=(θ/360)πr². So θ=120, r=9. So (120/360)π81 = (1/3)81π = 27π. So the answer should be 27π in².

Wait, but wait, maybe I made a mistake in the angle. Wait, a clock has 12 hours, so each hour mark is 30 degrees apart (360/12=30). At 4:00, the minute hand is at 12, hour at 4. So the number of intervals between 12 and 4 is 4, so 430=120 degrees. So the central angle is 120 degrees. Then sector area is (120/360)π9²=(1/3)81π=27π. So the correct option is 27π in².

Answer:

27π in² (corresponding to the option "27π in.²")