1. The number of hours of daylight D(t) in Halifax at day t can be approximated by the function
D(t) = a sin(bt + c) + d,
where t = 1 on January 1, and t = 365 on December 31. The period is one year, and the cycle starts on the vernal equinox, March 21, which has 12 hours of daylight (note that March 21 is day 80 in the year, since 31 + 28 + 21 = 80), reaches a maximum of about 15.3 hours at the summer solstice, and a minimum of about 8.7 hours at the winter solstice. Sketch the graph for the number of hours of daylight, and find a, b, c, and d to get the function. Then estimate the number of hours of daylight on your birthday.
2 a = 15.3 - 8.7
so
a = 3.3
d = (15.3 + 8.7)/2 = 12 logically enough
T = full period = 365
so sin 2 pi t/T = sin .01721 t
so b = .01721
sin (.01721 t - c) = 0 when t = 80
.01721 (80) - c = 0
c = 1.377
so in the end
D = 3.3 sin (.01721 t - 1.377) + 12
put in t = day of the year of your birthday. For example if your birthday is March 22 then t = 81. If birthday is Jan 2 then put in 2
Cool problem by the way, congratulate teacher !
Thank you so much Demon sir
To sketch the graph and find the values of a, b, c, and d, we can use the given information about the vernal equinox, summer solstice, and winter solstice.
1. Vernal Equinox (March 21, day 80): D(80) = 12 hours
2. Summer Solstice (approximately June 21): D(t) ≈ 15.3 hours
3. Winter Solstice (approximately December 21): D(t) ≈ 8.7 hours
Using this information, we can determine the values of a, b, c, and d.
a determines the amplitude of the function, so we can estimate it as the difference between the maximum and minimum values of daylight hours:
a ≈ (15.3 - 8.7) / 2 ≈ 3.3
The period is one year, which means the function completes one cycle in 365 days. This corresponds to a full period of 2π in the sin function. We can use this information to find b:
b = 2π / 365
The function starts on the vernal equinox, which is day 80, so we can set c such that
bt + c = 0 when t = 80.
Using the value of b found earlier:
(2π / 365) * 80 + c = 0
Solving for c gives us:
c ≈ - (2π / 365) * 80
Finally, to estimate d, we can substitute the values of a, b, c, and D(t) at the vernal equinox into the function:
a sin(bt + c) + d = 12
Substituting the known values:
3.3 sin((2π / 365) * 80 - (2π / 365) * 80) + d = 12
Simplifying:
3.3 sin(0) + d = 12
Since sin(0) = 0:
0 + d = 12
d = 12
Therefore, the function that approximates the number of hours of daylight in Halifax is:
D(t) = 3.3 sin((2π / 365) * t - (2π / 365) * 80) + 12
To estimate the number of hours of daylight on your birthday, substitute your birthday as t into the function:
D(t) = 3.3 sin((2π / 365) * t - (2π / 365) * 80) + 12
where t corresponds to your birthday.