Question

When do the hands of a clock coincide between 5 and 6 ? (b) \(5: 27: 16\) (a) \(5: 30\) (d) \(5: 28: 56\) (c) \(5: 32: 16\)

Short answer

a) 5:30 b) 5:27:16 c) 5:32:16 d) 5:28:56 Answer: None of the given options result in the hands of the clock coinciding between 5 and 6.

Step-by-step solution

Understand the position of clock hands

First, we need to understand how the positions of the clock hands work. For every hour, the hour hand moves 360 degrees, while the minute hand moves 12 times 360 = 4320 degrees (because the minute hand completes 12 cycles for every hour). Therefore, the movement of hour and minute hands is given by: Hour hand: 30 degrees per hour Minute hand: 360 degrees per hour (since it completes a full circle in an hour)

Find the position of the hour hand at 5

To find the position of the hour hand at 5, we simply multiply 5 (hours) by 30 (degrees per hour): Position of the hour hand = 5 * 30 = 150 degrees

Calculate position of hour hand at each given time

Now we need to find the position of the hour hand for each given time: (a) \(5:30\) - In \(30\) minutes, the hour hand would move half of the distance between 5 and 6, so the position is \(150 + 15 = 165\) degrees. (b) \(5:27:16\) - In \(27\) minutes and \(16\) seconds, the hour hand would move: \(\frac{27}{2}+\frac{16}{120}=\frac{816}{40}=20.4\) degrees past 5. So the position is \(150 + 20.4 = 170.4\) degrees. (c) \(5:32:16\) - In \(32\) minutes and \(16\) seconds, the hour hand would move: \(\frac{32}{2}+\frac{16}{120}=\frac{976}{40}=24.4\) degrees past 5. So the position is \(150 + 24.4 = 174.4\) degrees. (d) \(5:28:56\) - In \(28\) minutes and \(56\) seconds, the hour hand would move: \(\frac{28}{2}+\frac{56}{120}=\frac{840}{40}=21\) degrees past 5. So the position is \(150 + 21 = 171\) degrees.

Calculate position of minute hand at each given time

We need to find the position of the minute hand for each given time: (a) \(5:30\) - The minute hand is exactly at 6, so the position is: \(360 / 12 * 6 = 180\) degrees (b) \(5:27:16\) - The minute hand is at: \((360/60)* \left(27+\frac{16}{60}\right)=326.4\) degrees (c) \(5:32:16\) - The minute hand is at: \((360/60)* \left(32+\frac{16}{60}\right)=386.4\) degrees (but we need a number between 0 and 360, so we take this number modulus 360): \(386.4\pmod{360} = 26.4\) degrees (d) \(5:28:56\) - The minute hand is at: \((360/60)* \left(28+\frac{56}{60}\right)=347.2\) degrees

Compare positions of hour and minute hands

Now we compare the positions of the hour and minute hands for each given time: (a) \(5:30\) - Hour hand: \(165\) degrees, Minute hand: \(180\) degrees (do not coincide) (b) \(5:27:16\) - Hour hand: \(170.4\) degrees, Minute hand: \(326.4\) degrees (do not coincide) (c) \(5:32:16\) - Hour hand: \(174.4\) degrees, Minute hand: \(26.4\) degrees (do not coincide) (d) \(5:28:56\) - Hour hand: \(171\) degrees, Minute hand: \(347.2\) degrees (do not coincide)

Conclusion

None of the given options result in the hands of the clock coinciding between 5 and 6.

Key concepts

Time Calculation

Understanding time calculation on a clock is crucial for solving problems involving the coincidence of clock hands. Clock time consists of hours, minutes, and can even involve seconds to provide precise measurement. Each hour, the clock face divides time into 12 equal parts. This means every hour, the minute hand completes a full circle of 360 degrees, representing a new hour. Meanwhile, the hour hand moves 30 degrees per hour. Knowing these basic calculations helps one determine the exact position of each hand at any given time.

To calculate the position of the hour hand at any given time, you multiply the number of hours by 30, since each hour equates to 30 degrees. For minutes, it's simplified as *(time in hours)* times (degree per hour). Conversely, understanding the minute hand’s position requires you to first determine the percentage of a full circle it has completed. Given that each minute represents 6 degrees (360 divided by 60), you multiply the number of minutes by 6 to find out its position from the top of the clock or 12 o'clock position.

Degree Measurement

Degree measurement on a clock revolves around understanding how each hour at times is split into a certain number of degrees. Each number on a clock represents a 30-degree segment. This makes it essential to know how both clock hands 'claim' these degrees as they move.

The hour hand completes a full cycle of 360 degrees every 12 hours. Thus, translating to the hour hand moving 30 degrees each hour. For any additional minute after the hour, the movement is calculated proportionately. This involves multiplying the minutes past the hour by 0.5 degrees since every minute advances the hour hand by half a degree.

On the other hand, the minute hand, which operates over a 60-minute cycle, covers 6 degrees per minute. This detail is pivotal as these consistent steps allow one to map precisely where each hand resides at any point, ultimately aiding in determining when or if they coincide at any given time.

Minute and Hour Hand Movement

The movement of the clock's minute and hour hands plays a crucial role when dealing with clock angle problems. Recognizing how both hands move independently yet predictably gives insight into potential overlaps.

• The minute hand completes a full revolution every hour, moving 6 degrees per minute.
• The hour hand, conversely, travels slower, covering only 0.5 degrees per minute since it takes 12 hours to do a full circle.

This information is crucial for visualizing or even calculating the positions of each hand at any time. When calculating scenarios where the hands overlap or coincedence between 5 and 6 for instance, knowing that the minute hand is consistently gaining on the hour hand by 5.5 degrees per minute is vital. This pattern allows students to figure out how long it will take for both hands to meet (or not) within specified periods.

Problem Solving Techniques

When tackling clock angle problems such as determining when clock hands coincide, systematic techniques can simplify problem-solving. Begin by comprehending degrees traveled by both hands, determining their positions relative to the clock face.

• First, establish clear formulas, like degree formulas for hour or minute hand positions, that you'll use as benchmarks for calculations.
• Always multiply the known time (in hours or minutes) with the degree movement and take into account full cycles to get accurate positioning.
• Next, compare calculated positions of hands for specified times and see if they match (equal angles imply coincidence).

These step-by-step strategies prevent errors like missing degree recalculations and promote thorough understanding of the dynamic interplay of clock hands. Practicing these techniques can transform a complex problem into a more manageable task.