document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=317322>Question 317322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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.?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #227284 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=227284\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> clock chimes every hour on the hour.<BR>\n" );
document.write( "once at 1, twice at 2, etc.\r<BR>\n" );
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document.write( "clock takes 5 seconds to chime at 4.<BR>\n" );
document.write( "clock takes 15 seconds to chime at 10\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "This means the following.\r<BR>\n" );
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document.write( "At 1, the clock takes 0 seconds (negligible).<BR>\n" );
document.write( "At 2, the clock takes 1 and 2/3 seconds.<BR>\n" );
document.write( "At 3, the clock takes 3 and 1/3 seconds.<BR>\n" );
document.write( "At 4, the clock takes 5 seconds.<BR>\n" );
document.write( "At 5, the clock takes 6 and 2/3 seconds.<BR>\n" );
document.write( "At 6, the clock takes 8 and 1/3 seconds.<BR>\n" );
document.write( "at 7, the clock takes 10 seconds.<BR>\n" );
document.write( "at 8, the clock takes 11 and 2/3 seconds.<BR>\n" );
document.write( "at 9, the clock takes 13 and 1/3 seconds.<BR>\n" );
document.write( "at 10, the clock takes 15 seconds.<BR>\n" );
document.write( "at 11, the clock takes 16 and 2/3 seconds.<BR>\n" );
document.write( "at 12, the clock takes 18 and 1/3 second.\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "Let's assume the clock started as 12:59 pm.\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "The clock will pass through 1 through 12 for each revolution.\r<BR>\n" );
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document.write( "The clock will make 2 revolutions in a 24 hour period.\r<BR>\n" );
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document.write( "Each of the hours from 1 through 12 will have been chimed twice.\r<BR>\n" );
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document.write( "The sum of the seconds required for each revolution would be the sum of the seconds required for each chime from 1pm to 12pm..\r<BR>\n" );
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document.write( "This would be a total of 110 seconds for each revolutions which would be a total of 220 for 2 revolutions.\r<BR>\n" );
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document.write( "Time required for the clock to chime in a 24 hour period would be 220 seconds.\r<BR>\n" );
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document.write( "Detail Calculations are shown below:<BR>\n" );
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document.write( "event description        number of seconds\r\n" );
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document.write( "Clock strikes 1	         0.00000\r\n" );
document.write( "Clock strikes 2	         1.66667\r\n" );
document.write( "Clock strikes 3	         3.33333\r\n" );
document.write( "Clock strikes 4	         5.00000\r\n" );
document.write( "Clock strikes 5	         6.66667\r\n" );
document.write( "Clock strikes 6	         8.33333\r\n" );
document.write( "Clock strikes 7	        10.00000\r\n" );
document.write( "Clock strikes 8	        11.66667\r\n" );
document.write( "Clock strikes 9	        13.33333\r\n" );
document.write( "Clock strikes 10	15.00000\r\n" );
document.write( "Clock strikes 11	16.66667\r\n" );
document.write( "Clock strikes 12	18.33333\r\n" );
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document.write( "number of seconds for 1 revolution	110.00000\r\n" );
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document.write( "number of seconds for 2 revolutions	220.00000\r\n" );
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document.write( "This could have been modeled as the sum of an arithmetic series.\r<BR>\n" );
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document.write( "The formula for that would be:\r<BR>\n" );
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document.write( "Sn = n*(A1 + An)/2\r<BR>\n" );
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document.write( "To find An, we use the formula:\r<BR>\n" );
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document.write( "An = A1 + (n-1)d\r<BR>\n" );
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document.write( "d is equal to 1 and 2/3 seconds as we calculated above.<BR>\n" );
document.write( "A1 is equal to the number of seconds for the clock to chime at 1pm which is 0.<BR>\n" );
document.write( "n is equal to 12 because the clock chimes 12 times for each 12 hour period.<BR>\n" );
document.write( "n-1 is therefore equal to 11.\r<BR>\n" );
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document.write( "An = A1 + (n-1)d becomes An = 0 + 11*(1 and 2/3) which becomes 18.3333333 seconds.\r<BR>\n" );
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document.write( "Sum of Seconds required = Sn = n*(A1 + An)/2 becomes Sn = 12*(18.33333333)/2 which becomes:\r<BR>\n" );
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document.write( "Sn = 110\r<BR>\n" );
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document.write( "Two revolutions makes the total number of seconds equal to 220.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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