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A boy stands 1200 meters from a wall and claps his hands at regular time intervals. He adjusts the time interval between successive claps so that the echo from one clap coincides with the next clap. How many claps can he make in 6 minutes? Take the speed of sound as [tex]v=340 \, \text{m/s}[/tex].

Sagot :

GinnyAnswer
Sure, I'd be happy to help. Let's walk through the solution step-by-step:

1. Understanding the Problem:
- The boy is standing 1200 meters from a wall.
- He claps his hands at regular intervals such that each clap's echo coincides with the next clap.
- The problem asks for the number of claps he can make in 6 minutes.
- The speed of sound is given as 340 meters per second.

2. Convert the Total Time to Seconds:
- The total time given is 6 minutes.
- We know that 1 minute is 60 seconds.
- So, 6 minutes is [tex]\( 6 \times 60 \)[/tex] seconds.
- Therefore, [tex]\( 6 \times 60 = 360 \)[/tex] seconds.

3. Calculate the Distance the Echo Travels:
- The boy is standing 1200 meters from the wall.
- When the boy claps, the sound travels to the wall and back to him, which means it travels a distance of [tex]\( 2 \times 1200 \)[/tex] meters.
- Therefore, the total distance traveled by the echo is [tex]\( 2400 \)[/tex] meters.

4. Calculate the Time for One Echo:
- The speed of sound is given as 340 meters per second.
- We need to find out the time it takes for the sound to travel to the wall and back (i.e., 2400 meters).
- The time for one echo can be calculated using the formula: [tex]\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)[/tex].
- Substituting the values, we get [tex]\( \text{Time} = \frac{2400 \text{ meters}}{340 \text{ meters per second}} \)[/tex].
- Therefore, [tex]\( \text{Time} = 7.0588235294117645 \)[/tex] seconds (approximately 7.06 seconds).

5. Calculate the Number of Claps:
- We now need to find out how many times he can clap during the total time of 360 seconds.
- If each clap (including the echo round trip) takes approximately 7.06 seconds,
- Then, the number of claps he can make in 360 seconds can be found by dividing the total time by the time for one echo.
- Number of claps = [tex]\( \frac{360 \text{ seconds}}{7.0588235294117645 \text{ seconds per clap}} = 51 \)[/tex].

Thus, the boy can make 51 claps in 6 minutes if the time interval between claps is adjusted so that the echo coincides with the next clap.
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