Because Earth's rotation is gradually slowing, the length of each day increases: The day at the end of 1.0 century is 1.0 ms longer than the day at the start of the century. In 93 centuries, what is the total (in hours) of the daily increases in time (that is, the sum of the gain on the first day, the gain on the second day, etc.)?
To calculate the total increase in time over 93 centuries, we need to find the sum of the daily increases. Let's break down the problem step by step:
1. Find the increase in time for 1 century:
- Given that at the end of 1.0 century, the day is 1.0 ms longer than at the start of the century.
- There are 100 years in a century, and each year consists of 365.25 days (to account for leap years).
- To convert the increase in time from milliseconds (ms) to hours (hr), we need to divide by 3,600,000 (1 hour = 3,600,000 milliseconds).
- Therefore, the increase in time for 1 century is: (1.0 ms * 100 * 365.25) / 3,600,000 hr.
2. Calculate the total increase in time over 93 centuries:
- To find the sum of the gains on each day, we will multiply the increase for one century by the number of centuries.
- Total increase in time over 93 centuries = (increase in time for 1 century) * 93.
Let's perform the mathematical calculations:
Increase in time for 1 century = [(1.0 ms * 100 * 365.25) / 3,600,000] hr
Total increase in time over 93 centuries = (Increase in time for 1 century) * 93
Now, let's substitute the values into the equation:
Increase in time for 1 century = (1.0 * 100 * 365.25) / 3,600,000 hr
Total increase in time over 93 centuries = [(1.0 * 100 * 365.25) / 3,600,000] * 93 hr.
By calculating this expression, you'll find the total increase in time over 93 centuries in hours.