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

Explanation:

Step1: Calculate central - angle per part

The clock - face is divided into 12 equal parts. The total angle around a circle is $360^{\circ}$, so the central angle per part is $\frac{360^{\circ}}{12}=30^{\circ}$.

Step2: Analyze arc - length formula

The formula for the length of an arc of a circle is $s = r\theta$ (where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians). First, convert the angle from degrees to radians: $\theta$ (in radians)=$\frac{\pi}{180}\times\theta$ (in degrees).

Step3: Analyze the first statement

Between 11 and 2, there are 3 parts. The central angle $\theta_1 = 3\times30^{\circ}=90^{\circ}=\frac{\pi}{2}$ radians. The radius $r = 6$ inches, so the arc - length $s_1=r\theta_1=6\times\frac{\pi}{2}=3\pi$ inches. Between 7 and 10, there are 3 parts. The central angle $\theta_2 = 3\times30^{\circ}=90^{\circ}=\frac{\pi}{2}$ radians, and the arc - length $s_2=r\theta_2=6\times\frac{\pi}{2}=3\pi$ inches. So, the length of the minor arc between 11 and 2 is the same as the length of the minor arc between 7 and 10.

Step4: Analyze the second statement

Between 5 and 10, there are 5 parts. The central angle $\theta = 5\times30^{\circ}=150^{\circ}$. Convert to radians: $\theta=\frac{150\pi}{180}=\frac{5\pi}{6}$ radians. The radius $r = 6$ inches. Using the arc - length formula $s=r\theta=6\times\frac{5\pi}{6}=5\pi\approx5\times3.14 = 15.7$ inches.

Step5: Analyze the third statement

Between 2 and 4, there are 2 parts. The central angle $\theta=2\times30^{\circ}=60^{\circ}$.

Answer:

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