the figure above shows an angle $\theta = 0.812\pi$ in standard position and a circle with radius 3 in the xy-plane. what is the length of the minor arc of the circle from point u to point t?

(a) 0.850
(b) 2.551
(c) 5.101
(d) 7.653

Question

the figure above shows an angle $\theta = 0.812\pi$ in standard position and a circle with radius 3 in the xy-plane. what is the length of the minor arc of the circle from point u to point t?

(a) 0.850
(b) 2.551
(c) 5.101
(d) 7.653

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Unit Circle Trigonometry", "Standard Position Angles" ], "new_concepts": [ "Arc Length Formula", "Minor Arc Calculations" ], "current_concepts": [ "Unit Circle Trigonometry", "Standard Position Angles", "Arc Length Formula", "Minor Arc Calculations" ] } ## Identify the given parameters Using the *Standard Position Angles* and *Unit Circle Trigonometry* knowledge points $$ \begin{aligned} &r = 3\ &\theta = 0.812\pi \text{ rad} \end{aligned} $$ ## Determine the central angle for the minor arc Using the *Standard Position Angles* knowledge point $$ \begin{aligned} &\theta = 0.812\pi \approx 2.551 \text{ rad}\ &\theta \le \pi \implies \text{The angle } \theta \text{ directly defines the minor arc } UT. \end{aligned} $$ ## Calculate the minor arc length Using the *Arc Length Formula* knowledge point $$ \begin{aligned} &s = r \cdot \theta\ &s = 3 \cdot (0.812\pi)\ &s = 2.436\pi \approx 7.653 \end{aligned} $$

0.850

2.551

5.101

7.653

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Arc Length Formula" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given parameters

Determine the central angle for the minor arc

Calculate the minor arc length

Answer: