# Precalculus II

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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.

• Precalculus II -

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

• Precalculus II -

Cool problem by the way, congratulate teacher !

• Precalculus II -

Thank you so much Demon sir

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