Hello, I just have a basic unit conversion problem I just can\'tseem to get righ

Question

Hello, I just have a basic unit conversion problem I just can'tseem to get right. The problem:

Because Earth's rotation is gradually slowing, the length ofeach day increases: The day at the end of 1.0 century is 1.0 mslonger than the day at the start of the century. In 27 centuries,what is the total (in hours) of the daily increases in time (thatis, the sum of the gain on the first day, the gain on the secondday, etc.)?

Now I don't understand how I am supposed to take the 1ms intoaccount for this problem, and if they want the full gain of everyday or just the 1ms per century gain. I am a little lost ifsomeone would be able to help I would appreciate it. Thankyou.

Explanation / Answer

Gain in time for 1 yr = 1 * 10E-5 = Tyr Gain in time for 1 day = Tyr / 365 = 2.73972 *10E-8 sec Suppose a is the time elapsed in the firstday    Then in 2 days the time elapsed is 1 + (1+ x) After 3 days the time elapsed is 1 + (1 + x) + (1 +2x) This is an arithmetic progression with the sum S = n / 2[2 a + (n - 1) x] The term n / 2 * 2 a is just the number of days n *a So the sum we are interested in is S = n / 2 (n - 1)x In 2700 years we have 2700 * 365 = 985500 days This can be refined a little for leap years (25 per century *27 centuries) = 675 days (Actually centuries not divisible by 400 skip a leap day -1700, 1800, 1900, etc. so we'll just call this 670) Using 986170 as the number of days The sum becomes S (sec) = 986170 / 2 * 986169 * 2.73972* 10E-8 = 13322 sec The time in hours is then t = 3.7 hours

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