Question

the face of a clock is divided into 12 equal parts. the radius of the clock face is 10 inches. assume the hands of the clock will form a central angle. which statements about the clock are accurate? select three options. the central angle formed when one hand points at 1 and the other hand points at 3 is 30°. the circumference of the clock is approximately 62.8 inches. the minor arc measure when one hand points at 12 and the other hand points at 4 is 120°. the length of the major arc between 3 and 10 is approximately 31.4 inches. the length of the minor arc between 6 and 7 is approximately 5.2 inches.

Explanation:

Step1: Calculate central angle per part

The total angle in 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 angle between 1 and 3: There are \(3 - 1=2\) parts. The angle is \(2\times30^{\circ} = 60^{\circ}

eq30^{\circ}\).

Step2: Calculate circumference

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(r = 10\) inches, \(C=2\times3.14\times10=62.8\) inches.

Step3: Calculate minor - arc measure between 12 and 4

There are 4 parts between 12 and 4. The angle (minor - arc measure) is \(4\times30^{\circ}=120^{\circ}\).

Step4: Calculate major - arc length between 3 and 10

The number of parts between 3 and 10 (major - arc) is \(12-(10 - 3)=5\) parts. The angle is \(5\times30^{\circ}=150^{\circ}\). The formula for arc length is \(L=\frac{\theta}{360^{\circ}}\times2\pi r\). \(L=\frac{150^{\circ}}{360^{\circ}}\times2\times3.14\times10=\frac{5}{12}\times62.8\approx26.2
eq31.4\) inches.

Step5: Calculate minor - arc length between 6 and 7

There is 1 part between 6 and 7. The angle is \(30^{\circ}\). Using the arc - length formula \(L=\frac{\theta}{360^{\circ}}\times2\pi r\), \(L=\frac{30^{\circ}}{360^{\circ}}\times2\times3.14\times10=\frac{1}{12}\times62.8\approx5.2\) inches.

Answer:

The circumference of the clock is approximately 62.8 inches; The minor arc measure when one hand points at 12 and the other hand points at 4 is \(120^{\circ}\); The length of the minor arc between 6 and 7 is approximately 5.2 inches.