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 $30^\circ$.
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 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.

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 $30^\circ$.
  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 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.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Arc Measure", "Circle Geometry" ], "new_concepts": [ "Arc Length Formula" ], "current_concepts": [ "Arc Measure", "Circle Geometry", "Arc Length Formula" ] } ## Calculate basic clock parameters $$ \begin{aligned} &r = 10\text{ inches} \ &C = 2\pi r = 20\pi \approx 62.83\text{ inches} \ &\text{Angle per hour unit} = \frac{360^\circ}{12} = 30^\circ \ &\text{Arc length per hour unit} = \frac{C}{12} = \frac{20\pi}{12} = \frac{5\pi}{3} \approx 5.24\text{ inches} \end{aligned} $$ ## Evaluate each statement $$ \begin{aligned} &\text{Statement 1 (1 to 3): } \text{Angle} = 2 \times 30^\circ = 60^\circ eq 30^\circ \quad (\text{Incorrect}) \ &\text{Statement 2 (Circumference): } C \approx 62.8\text{ inches} \quad (\text{Correct}) \ &\text{Statement 3 (12 to 4): } \text{Minor arc measure} = 4 \times 30^\circ = 120^\circ \quad (\text{Correct}) \ &\text{Statement 4 (3 to 10): } \text{Major arc units} = 12 - 7 = 5\text{ units (or minor is 5, major is 7)} \ &\quad \text{Major arc length} = 7 \times 5.24 = 36.68\text{ inches} eq 31.4\text{ inches} \quad (\text{Incorrect}) \ &\text{Statement 5 (6 to 7): } \text{Minor arc length} = 1 \times 5.24 \approx 5.2\text{ inches} \quad (\text{Correct}) \end{aligned} $$ ## Select the three accurate options $$ \text{Correct statements: Statement 2, Statement 3, and Statement 5} $$

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.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circle Geometry" ] }

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Question

Explanation:

Calculate basic clock parameters

Evaluate each statement

Select the three accurate options

Answer: