periodic and sinusoidal functions graded assignment

03 evaluate
each part a = 10 points, part b = 10 points, part c = 5 points

1. the hour hand of a standard clock moves with an angular velocity of $4\pi$ radians per day.

a) how many radians does the hour hand rotate through in 30 minutes?

b) determine how many hours it will take for the hour hand to rotate $5\pi/2$ radians.

c) suppose we observe the hour hand starting at 2pm. when it completes a rotation of $17\pi$ radians what time will it be? specify the time and am or pm.

Question

periodic and sinusoidal functions graded assignment

03 evaluate
each part a = 10 points, part b = 10 points, part c = 5 points

1. the hour hand of a standard clock moves with an angular velocity of $4\pi$ radians per day.

a) how many radians does the hour hand rotate through in 30 minutes?

b) determine how many hours it will take for the hour hand to rotate $5\pi/2$ radians.

c) suppose we observe the hour hand starting at 2pm. when it completes a rotation of $17\pi$ radians what time will it be? specify the time and am or pm.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Angular Velocity", "Unit Conversion" ], "new_concepts": [ "Clock Rotation Calculations" ], "current_concepts": [ "Angular Velocity", "Unit Conversion", "Clock Rotation Calculations" ] } ## Convert time to days for rotation Using the *Unit Conversion* and *Angular Velocity* knowledge points $$ \begin{aligned} &t = 30\text{ minutes} = 0.5\text{ hours}\ &\text{Since } 1\text{ day} = 24\text{ hours}:\ &t = \frac{0.5}{24}\text{ days} = \frac{1}{48}\text{ days}\ &\theta = \omega \cdot t = 4\pi \cdot \frac{1}{48} = \frac{\pi}{12}\text{ radians} \end{aligned} $$ ## Solve for time given rotation angle Using the *Angular Velocity* knowledge point $$ \begin{aligned} &\theta = \frac{5\pi}{2}\text{ radians}\ &t_{\text{days}} = \frac{\theta}{\omega} = \frac{\frac{5\pi}{2}}{4\pi} = \frac{5}{8}\text{ days}\ &t_{\text{hours}} = \frac{5}{8}\text{ days} \times 24\text{ hours/day} = 15\text{ hours} \end{aligned} $$ ## Calculate final time after rotation Using the *Angular Velocity* and *Unit Conversion* knowledge points $$ \begin{aligned} &\theta = 17\pi\text{ radians}\ &t_{\text{days}} = \frac{17\pi}{4\pi} = 4.25\text{ days}\ &t_{\text{hours}} = 4.25\text{ days} \times 24\text{ hours/day} = 102\text{ hours}\ &\text{Since } 102 = 4 \times 24 + 6:\ &102\text{ hours} = 4\text{ full days} + 6\text{ hours}\ &\text{Starting at 2:00 PM} + 6\text{ hours} = \text{8:00 PM} \end{aligned} $$ ## Question 1a The hour hand rotates through $\frac{\pi}{12}$ radians in 30 minutes. ## Question 1b It will take 15 hours for the hour hand to rotate $\frac{5\pi}{2}$ radians. ## Question 1c The time will be 8:00 PM. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Angular Velocity" ] }

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QUESTION IMAGE

Question

Explanation:

Convert time to days for rotation

Solve for time given rotation angle

Calculate final time after rotation

Answer:

Question 1a

Question 1b

Question 1c