Problem Solving (Word Problem)

Question

The number of hours of daylight on the summer solstice (the 172nd day of the year) is 15.3 and the number of hours of daylight on the winter solstice (the 355th day of the year) is 9.1. Complete the function below to model this data.

y= [?]cos(2pi/[?](x-[?]))+[?]

Answer 1

Defining y == hours of sunlight, x == day or the year

rewrite the equation with variables for simplicity:

y = A cos(2π /[B(x-C)] + D

D will be the average number of hours of sunlight, while the cosine function is the adjustment to the average for a given day. In this model, a complete cycle occurs every 366 days (the time between 172 and 355 is 183, 2X that is 366).

Find D by averaging the max and min values of y, D = (15,3 + 9.1)/2 = 24.4/2 = 12.2. Find A by finding the difference between the average and the high (and low) values A = 15.3 - 12.2 = 3.1. The maximum value of cosine is +1 which occurs at 0 and multiples of 2π; the minimum value of cosine is -1 which occurs at odd multiples of π. So want the cosine value at 2π on day 172 and want the cosine value at π on day 355.

so: 2π = 2π/(B(172-C)) --> B = 1/(172 - C)

and: π = 2π/(B(355-C)) --> B = 2/(355 - C)

Combining these: 355 - C = 2 ( 172 - C) = 344 -2C --> C = 344 - 355 = -11

Use one of the other equations to solve for B: B = 1/(172 - (-11)) = 1/183

Now you can write the equation, replacing variables with the calculated values