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:

Step1: Calculate central angle per part

The total central angle of a circle is \(360^{\circ}\). Since the clock is divided into 12 equal parts, the central angle per part is \(\frac{360^{\circ}}{12}=30^{\circ}\).
For the hands at 2 and 4, the number of parts between them is \(4 - 2=2\). So the central angle is \(2\times30^{\circ}=60^{\circ}\).

Step2: Calculate the circumference

The formula for the circumference of a circle is \(C = 2\pi r\), where \(r = 6\) inches. So \(C=2\pi\times6=12\pi\approx12\times3.14 = 37.68\) inches (not \(19\) inches).

Step3: Calculate arc length

The formula for arc length is \(s=r\theta\) (where \(\theta\) is in radians). For hands at 5 and 10, the number of parts between them is \(10 - 5 = 5\). The central angle \(\theta=5\times30^{\circ}=150^{\circ}=\frac{150\pi}{180}=\frac{5\pi}{6}\) radians. With \(r = 6\) inches, \(s=6\times\frac{5\pi}{6}=5\pi\approx5\times3.14 = 15.7\) inches.

Step4: Calculate minor arc measure

For hands at 1 and 9, the number of parts between them (minor arc) is \(9 - 1=8\) (but \(12-(9 - 1)=4\) parts for minor arc). The central angle is \(4\times30^{\circ}=120^{\circ}\) (not \(150^{\circ}\)).

Step5: Check arc lengths

For the arc between 11 and 2: number of parts \(2-(11 - 12)=3\) (since \(11\) is \(11\) and \(2\) with wrap - around). For the arc between 7 and 10: number of parts \(10 - 7=3\). Since \(s = r\theta\) and \(r\) is the same (\(r = 6\)) and \(\theta\) (in terms of number of parts \(\times30^{\circ}\)) is the same (\(3\times30^{\circ}\)), the arc lengths are the same.

Answer:

• The central angle measure when one hand points at 2 and the other points at 4 is \(60^{\circ}\).
• 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.