Question

the face of a clock is divided into 12 equal parts. the radius of the clock face is 6 inches. assume the hands of the clock will form a central angle.
which statements about the clock are accurate? check all that apply.

• the central angle measure when one hand points at 2 and the other points at 4 is 60°.
• the circumference of the clock is about 19 in.
• with one hand at 5 and the other at 10, the minor arc formed by the hands is about 15.7 in.
• the minor arc measure when one hand points at 1 and the other hand points at 9 is 150°.
• the length of the minor arc between 11 and 2 is the same as the length of the minor arc between 7 and 10.

Explanation:

First, recall that a clock is a circle (360°) divided into 12 equal parts. So each hour mark represents an angle of \( \frac{360^\circ}{12} = 30^\circ \). The circumference of a circle is \( C = 2\pi r \), and the length of an arc is \( s = r\theta \) (where \( \theta \) is in radians) or \( s = \frac{\theta}{360^\circ} \times 2\pi r \) (where \( \theta \) is in degrees). The radius \( r = 6 \) inches.

Statement 1: Central angle at 2 and 4

The number of hours between 2 and 4 is \( 4 - 2 = 2 \). Each hour is \( 30^\circ \), so the central angle is \( 2 \times 30^\circ = 60^\circ \). This is accurate.

Statement 2: Circumference of the clock

Circumference \( C = 2\pi r = 2\pi(6) = 12\pi \approx 37.7 \) inches (not 19). This is inaccurate.

Statement 3: Minor arc at 5 and 10

Number of hours between 5 and 10: \( 10 - 5 = 5 \). Central angle \( \theta = 5 \times 30^\circ = 150^\circ \). Arc length \( s = \frac{150^\circ}{360^\circ} \times 2\pi(6) = \frac{5}{12} \times 12\pi = 5\pi \approx 15.7 \) inches. This is accurate.

Statement 4: Minor arc at 1 and 9

Number of hours between 1 and 9: \( 9 - 1 = 8 \), but since it's a minor arc, we take the smaller angle. The total hours in a circle is 12, so the smaller number of hours is \( 12 - 8 = 4 \)? Wait, no: 1 to 9 is 8 hours, but the minor arc is the smaller one. Wait, 1 to 9: clockwise, it's 8 hours, counterclockwise, it's 4 hours? Wait, no, 1 to 9: 9 - 1 = 8, but the minor arc is the smaller angle. Wait, 360° - (8×30°) = 360° - 240° = 120°? Wait, no, I made a mistake. Wait, 1 to 9: the number of hours between them is 8, but the minor arc is the smaller angle, so 12 - 8 = 4? No, that's not right. Wait, 1 to 9: from 1 to 9, moving clockwise, it's 8 hours (8×30°=240°), but the minor arc is the smaller angle, so 360° - 240° = 120°? Wait, no, that's incorrect. Wait, 1 to 9: 9 - 1 = 8, but the minor arc is the smaller of the two arcs. Wait, 8×30°=240°, which is more than 180°, so the minor arc is 360° - 240° = 120°, not 150°. Wait, so the central angle for the minor arc is 120°, not 150°. So this statement is inaccurate. Wait, no, wait: 1 to 9: 9 - 1 = 8, but the minor arc is the smaller angle, so 12 - 8 = 4? No, that's not how it works. Wait, the minor arc is the shortest path between the two points. So from 1 to 9: clockwise, 8 hours (240°), counterclockwise, 4 hours (120°). So the minor arc is 120°, not 150°. So this statement is inaccurate. Wait, but let's recalculate. Wait, 5 to 10: 5 hours, 150°, which is less than 180°, so that's the minor arc. But 1 to 9: 8 hours, which is more than 180°, so the minor arc is 4 hours (120°). So the central angle for the minor arc at 1 and 9 is 120°, not 150°. So this statement is inaccurate.

Wait, I think I messed up. Let's correct: the minor arc is the smaller of the two possible arcs between two points. So between 1 and 9: the number of hours is 8 (clockwise) or 4 (counterclockwise). Since 4 < 8, the minor arc is 4 hours, so central angle 4×30°=120°, not 150°. So statement 4 is inaccurate.

Statement 5: Minor arc between 11 and 2, and 7 and 10

Number of hours between 11 and 2: 11 to 12 is 1, 12 to 2 is 2, total 3 hours. Between 7 and 10: 10 - 7 = 3 hours. So both have 3 hours, so the central angle is \( 3 \times 30^\circ = 90^\circ \). Arc length \( s = \frac{90^\circ}{360^\circ} \times 2\pi(6) = \frac{1}{4} \times 12\pi = 3\pi \) inches for both. So their lengths are the same. This is accurate.

Wait, let's recheck each statement:

1. Accurate.
2. Inaccurate.
3. Accurate (arc length ≈15.7).
4. Inaccurate (minor arc angle is 120°, not 150°).
5. Accurate (both 3 hours apart, same arc length).

Wait, let's recheck statement 4: 1 to 9. The number of hours between 1 and 9: 9 - 1 = 8. The minor arc is the smaller angle, so 12 - 8 = 4? No, 8 hours is 240°, 4 hours is 120°, so minor arc is 120°, so the central angle is 120°, not 150°. So statement 4 is wrong.

Statement 3: 5 to 10: 5 hours, 150°, arc length \( \frac{150}{360} \times 12\pi = 5\pi ≈15.7 \), correct.

Statement 5: 11 to 2: 11 to 12 (1), 12 to 2 (2), total 3 hours. 7 to 10: 10 - 7 = 3 hours. So both 3 hours, same arc length. Correct.

So the accurate statements are:

- The central angle measure when one hand points at 2 and the other points at 4 is 60°.
- With one hand at 5 and the other at 10, the minor arc formed by the hands is about 15.7 in.
- The length of the minor arc between 11 and 2 is the same as the length of the minor arc between 7 and 10.

Wait, let's confirm statement 4 again. 1 to 9: the angle between 1 and 9. If we go clockwise from 1 to 9, it's 8 hours (8×30=240°), counterclockwise, it's 4 hours (4×30=120°). So the minor arc is 120°, so the central angle is 120°, not 150°. So statement 4 is incorrect.

So the correct statements are the first, third, and fifth.

Answer:

• The central angle measure when one hand points at 2 and the other points at 4 is 60°.
• With one hand at 5 and the other at 10, the minor arc formed by the hands is about 15.7 in.
• The length of the minor arc between 11 and 2 is the same as the length of the minor arc between 7 and 10.