Question

trigonometric function question:
In 2001, Windsor, Ontario will receive its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on December 21.

a) Due to the earth's revolution about the sun, the hours of daylight function is periodic. Determine an equation that can model the hours of daylight function for Windsor, Ontario.

b) On what day(s) can Windsor expect 13.5 hours of sunlight?

(I'm pretty sure that you use this equation y= a sin[b(x-c)]+d )

Answer 1

correct you will use y = a sin(b(x-c)) + d
The range of sin is from 1 to -1 or 2 which is max - min
So to find range of our function 15.28 - 9.08 = 6.2
Therefore our function goes from -3.1 to 3.1 with middle value of 0.
-> a = 3.1
Now what do we have to add to 3.1 to get 15.28 that is our shift up
-> d = 15.28 - 3.1 = 12.18
Ok i am assuming using degrees and that num of days in a year are 360 so i day corresponds loosely to one degree. This way the period of our sin wave does not change. ->b = 360/360 = 1
For the left/right shift of our graph think about the when sin is at max/min
-> sin90 = 1 and sin270 = -1
june 21 = day170 (30 day/month, jan.1 is day 0)
dec 21 = day 350
-> c = 170 - 90 or 350 - 270 = 80
->y = 3.1sin(x-80) +12.18
this is an approximation but hopefully gives you an idea. to find which day given hours of sunlight. solve equation for x
->(y-12.18)/3.1 = sin(x-80) ..

Answer 2

thanks so much for the help!

Answer 3

your welcome :)