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Question 317322: The clock in the bell tower at Webster College chimes every hour on the hour: once at 1 o’clock, twice at 2 o’clock, and so on. The clock takes 5 seconds to chime at 4 o’clock and 15 seconds to chime at 10 o’clock. The time needed to chime 1 o’clock is negligible. What is the total number of seconds needed for the clock to do all of its chiming in a 24-hour period starting at 1 P.M.?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! clock chimes every hour on the hour.
once at 1, twice at 2, etc.
clock takes 5 seconds to chime at 4.
clock takes 15 seconds to chime at 10
If the clock takes 5 seconds to chime at 4 and 15 seconds to chime at 10, then the clock increases the duration of its chime by 1 and 2/3 seconds each time.
This means the following.
At 1, the clock takes 0 seconds (negligible).
At 2, the clock takes 1 and 2/3 seconds.
At 3, the clock takes 3 and 1/3 seconds.
At 4, the clock takes 5 seconds.
At 5, the clock takes 6 and 2/3 seconds.
At 6, the clock takes 8 and 1/3 seconds.
at 7, the clock takes 10 seconds.
at 8, the clock takes 11 and 2/3 seconds.
at 9, the clock takes 13 and 1/3 seconds.
at 10, the clock takes 15 seconds.
at 11, the clock takes 16 and 2/3 seconds.
at 12, the clock takes 18 and 1/3 second.
In a 24 hour period, if the clock chimes starts chiming at 1pm, then it must have started some time before 1pm and after 12pm.
Let's assume the clock started as 12:59 pm.
In a 24 hour period it will start at 12:59 pm, pass through 12:59 pm 1 time and stop at 12:59pm the second time.
The clock will pass through 1 through 12 for each revolution.
The clock will make 2 revolutions in a 24 hour period.
Each of the hours from 1 through 12 will have been chimed twice.
The sum of the seconds required for each revolution would be the sum of the seconds required for each chime from 1pm to 12pm..
This would be a total of 110 seconds for each revolutions which would be a total of 220 for 2 revolutions.
Time required for the clock to chime in a 24 hour period would be 220 seconds.
Detail Calculations are shown below:
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event description number of seconds
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Clock strikes 1 0.00000
Clock strikes 2 1.66667
Clock strikes 3 3.33333
Clock strikes 4 5.00000
Clock strikes 5 6.66667
Clock strikes 6 8.33333
Clock strikes 7 10.00000
Clock strikes 8 11.66667
Clock strikes 9 13.33333
Clock strikes 10 15.00000
Clock strikes 11 16.66667
Clock strikes 12 18.33333
number of seconds for 1 revolution 110.00000
number of seconds for 2 revolutions 220.00000
This could have been modeled as the sum of an arithmetic series.
The formula for that would be:
Sn = n*(A1 + An)/2
To find An, we use the formula:
An = A1 + (n-1)d
d is equal to 1 and 2/3 seconds as we calculated above.
A1 is equal to the number of seconds for the clock to chime at 1pm which is 0.
n is equal to 12 because the clock chimes 12 times for each 12 hour period.
n-1 is therefore equal to 11.
An = A1 + (n-1)d becomes An = 0 + 11*(1 and 2/3) which becomes 18.3333333 seconds.
Sum of Seconds required = Sn = n*(A1 + An)/2 becomes Sn = 12*(18.33333333)/2 which becomes:
Sn = 110
Two revolutions makes the total number of seconds equal to 220.
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