The number of daylight hours in a day is harmonic. Suppose that in a particular location, the shortest day of the year has 7 hours of daylight and the longest day of the year has 18 hours. Then, we can model its motion with the function N=Asin(Bt) + C where t is expressed in days and A, B > 0. Find A and B, using 365 days for one year.
I need to find what A, B, and C equal. Thanks.
In the equation:
N=Asin(Bt) + C
A represents half of the difference between the maximum and minimum, sometimes called amplitude.
B is a multiplitive factor to ensure that the cycle of sine (2π) fits into the physical cycle, in this problem, 365 days.
Since the amplitude of daylight hours is (18-7)/2=5.5, we obtain A=5.5.
We need a value of B such that when t=0 and t=365, Bt becomes 0 and 2π.
So 365B = 2π, or B=2π/365.
To find the values of A, B, and C in the equation N = Asin(Bt) + C, we need to use the given information about the shortest and longest day of the year.
Let's start by considering the shortest day, which has 7 hours of daylight. We can substitute the values into the equation:
7 = Asin(B * shortest day) + C
Next, let's consider the longest day, which has 18 hours of daylight:
18 = Asin(B * longest day) + C
Now, let's use the fact that there are 365 days in a year. Since we want to model the entire year, we need to find the period of the function, which is the time it takes for the function to complete one full cycle. In this case, the period is 365 days. The period of a sine function is given by 2π/B, where B is the coefficient of t in the equation.
So, 2π/B = 365
To solve for B, we can rearrange the equation:
B = 2π/365
Now that we have the value of B, we can go back to the previous equations:
7 = Asin(B * shortest day) + C
18 = Asin(B * longest day) + C
Remember that A and B are positive constants, so the values of N will vary between A+C and -A+C.
Using these two equations, we can solve for A and C simultaneously. Subtracting the two equations gives us:
18 - 7 = Asin(B * longest day) - Asin(B * shortest day)
11 = A[sin(B * longest day) - sin(B * shortest day)]
Now, we can divide both sides by sin(B * longest day) - sin(B * shortest day):
11 / (sin(B * longest day) - sin(B * shortest day)) = A
Finally, substituting the value of B we found earlier, we can calculate the value of A. Once we have A, we can substitute it back into one of the original equations to solve for C.
Note: The exact values of A, B, and C will depend on the specific location under consideration.