Math Problem Statement
the angle between the hour and minute hand is 197 time is between 8 and 9 pm
Solution
To determine the exact time between 8:00 PM and 9:00 PM where the angle between the hour and minute hands is , we can solve this mathematically.
Step-by-step Solution:
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Define Positions of the Hands:
- The hour hand moves per minute ().
- The minute hand moves per minute ().
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Relative Motion Between the Hands: At 8:00 PM, the hour hand starts at from the top (12 o'clock position): \text{Position of the hour hand at time t: } H(t) = 240 + 0.5t The minute hand starts at and moves per minute: \text{Position of the minute hand at time t: } M(t) = 6t
The absolute difference between the two hands at time is:
Simplify the expression:
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Set the Equation: The angle is given as :
Solve for :
- Case 1:
- Case 2:
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Interpret the Results: Since the time must be between 8:00 and 9:00 PM:
- minutes after 8:00 PM corresponds to (approximately ).
- minutes after 8:00 PM is beyond 9:00 PM, so it is invalid.
Thus, the time is approximately 8:07:49 PM.
Would you like details on how these calculations were derived or have other questions? Here are some related questions to deepen understanding:
- How do we calculate the speed of each hand of a clock?
- Why do we consider absolute differences for angles between the hands?
- How can we determine the time when the angle is exactly 90 degrees?
- What are the mathematical principles behind clock problems?
- How does this calculation change if the time range shifts (e.g., between 7 and 8 PM)?
Tip: To quickly estimate clock angles, remember the relative speeds: per minute for the two hands.
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Math Problem Analysis
Mathematical Concepts
Angle between clock hands
Linear equations
Motion of clock hands
Formulas
Position of the hour hand: H(t) = 240 + 0.5t
Position of the minute hand: M(t) = 6t
Angle between the hands: |M(t) - H(t)| = |6t - (240 + 0.5t)|
Theorems
Principle of relative motion in circular paths
Suitable Grade Level
Grades 8-10