) The number of hours, H
, of daylight in Madrid as a function of date is approximated by the formula
H=12+2.4sin(0.0172(t−80)),
where t
is the number of days since the start of the year. (We can think of t=0
as the stroke of midnight on Dec. 31/Jan 1; thus, January falls between t=0
and t=31
, February falls between t=31
and t=59
, etc.).
Find the average number of hours of daylight in Madrid (assuming in each case that it is not a leap year):
A.) in December: average hours =
B.) in July: average hours =
C.) over a year: average hours =
A.) To find the average number of hours of daylight in Madrid in December, we need to calculate the total number of daylight hours in December and divide by the number of days in December. December falls between t = 365-31 = 334 and t = 365, so we need to plug in values of t between 334 and 365 into the formula given:
H = 12 + 2.4sin(0.0172(t-80))
We can use a calculator or software to do this, or we can use a spreadsheet to generate a table of values. For example, we can create a table with the following columns:
- Date (starting from December 1 and ending on December 31)
- t (corresponding to each date, calculated as t = 334 + date - 1 since December 1 is t = 334)
- H (calculated using the formula)
Using this table, we can then calculate the total number of daylight hours by summing the H values, and then divide by 31 (the number of days in December). Alternatively, we can use software or a calculator to integrate the function over the range t = 334 to t = 365, and then divide by 31. Either way, we get:
Average hours = (1/31) * ∫334^365 (12 + 2.4sin(0.0172(t-80))) dt
≈ 9.06 hours
Note that this is an approximation, since the function is only an approximation of the actual daylight hours. Also, the actual number of hours of daylight will vary depending on the year and other factors.
B.) Following the same method, we can find the average number of hours of daylight in Madrid in July. July falls between t = 180 and t = 210, so we need to integrate the function over this range:
Average hours = (1/31) * ∫180^210 (12 + 2.4sin(0.0172(t-80))) dt
≈ 14.52 hours
C.) To find the average number of hours of daylight in Madrid over a year, we need to calculate the total number of daylight hours over the whole year and divide by the number of days in a year (365 or 366, depending on if it's a leap year). We can either estimate this by using the formula for each day of the year and summing the values, or we can use software or a calculator to integrate the function over the entire year:
Average hours = (1/365) * ∫0^365 (12 + 2.4sin(0.0172(t-80))) dt
≈ 12.06 hours
Again, this is just an approximation and the actual number of daylight hours will vary from year to year.