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.
