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 197197^\circ, we can solve this mathematically.

Step-by-step Solution:

  1. Define Positions of the Hands:

    • The hour hand moves 0.50.5^\circ per minute (360/720minutes360^\circ / 720 \, \text{minutes}).
    • The minute hand moves 66^\circ per minute (360/60minutes360^\circ / 60 \, \text{minutes}).
  2. Relative Motion Between the Hands: At 8:00 PM, the hour hand starts at 240240^\circ 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 00^\circ and moves 66^\circ per minute: \text{Position of the minute hand at time t: } M(t) = 6t

    The absolute difference between the two hands at time tt is: Angle: M(t)H(t)=6t(240+0.5t)\text{Angle: } |M(t) - H(t)| = |6t - (240 + 0.5t)|

    Simplify the expression: 6t2400.5t=5.5t240|6t - 240 - 0.5t| = |5.5t - 240|

  3. Set the Equation: The angle is given as 197197^\circ: 5.5t240=197|5.5t - 240| = 197

    Solve for tt:

    • Case 1: 5.5t240=1975.5t - 240 = 197 5.5t=437t=4375.579.45minutes5.5t = 437 \quad \Rightarrow \quad t = \frac{437}{5.5} \approx 79.45 \, \text{minutes}
    • Case 2: 5.5t240=1975.5t - 240 = -197 5.5t=43t=435.57.82minutes5.5t = 43 \quad \Rightarrow \quad t = \frac{43}{5.5} \approx 7.82 \, \text{minutes}
  4. Interpret the Results: Since the time must be between 8:00 and 9:00 PM:

    • t=7.82t = 7.82 minutes after 8:00 PM corresponds to 8:07.828:07.82 (approximately 8:088:08).
    • t=79.45t = 79.45 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:

  1. How do we calculate the speed of each hand of a clock?
  2. Why do we consider absolute differences for angles between the hands?
  3. How can we determine the time when the angle is exactly 90 degrees?
  4. What are the mathematical principles behind clock problems?
  5. 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: 5.55.5^\circ 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