A city averages 14 hours of daylight in June, 10 in December, 12 in both March and September. Assume that the number of hours of daylight varies sinusoidally over a period of one year. Write an expression for n, the number of hours of daylight, as a cosine function of t. Let t be in months and t=0 correspond to the month of January.

My work:
y=2cos((pi/6(x-d))+12

Ok how would I get the phase shift? Thanks in advance

treat one of the data values as input into your partial equation.

e.g. 14 hours in June --- y = 14, x = 5

14 = 2cos((pi/6(5-d))+12
2 = 2cos((pi/6(5-d))
1 = cos((pi/6(5-d))
so pi/6(5-d) = 2pi , because cos 2pi = 1
which solves for d = -7

your equation is

y=2cos(pi/6)(x+7) + 12

test it for one other given value.
e.g. Dec --> x = 11
y = 2cos(pi/6)18 + 12
= 2cos(3pi) + 12
= 2(-1) + 12
= 10 , as given

how r u getting pi?

Well, it seems like you're off to a good start with the formula for a cosine function! However, there is a small mistake. The phase shift of the function represents the horizontal displacement of the graph.

To find the phase shift, you can observe that in June, the city averages 14 hours of daylight, which is the maximum value. Since June corresponds to the 6th month of the year, the maximum value of the cosine function should occur at t = 6. Therefore, the phase shift (d) should be 6 months.

So the corrected expression for n, the number of hours of daylight, as a cosine function of t, would be:

n = 2cos((π/6)(t-6)) + 12

Keep up the good work, and don't hesitate to reach out if you have any further questions!

To determine the phase shift in this scenario, you need to find the value of "d" in the equation. The phase shift represents the horizontal shift (left or right) of the cosine function.

To find the phase shift, you can use the given information about the number of hours of daylight for different months.

Let's analyze the information given:
- In June, the city averages 14 hours of daylight.
- In December, the city averages 10 hours of daylight.

Since June has the maximum number of daylight hours (14), this corresponds to the maximum value of the cosine function. In the cosine function, the maximum occurs at the phase angle 0 (or 2π).

Therefore, for June (t = 6 months), we have: cos(2π) = 14.

Similarly, for December (t = 12 months), we have: cos(4π) = 10.

Now, we can set up the equation to solve for the phase shift (d).

Using the cosine function expression: y = 2cos((π/6(t-d)) + 12,

Substituting t = 6 and y = 14 into the equation:
14 = 2cos((π/6(6 - d)) + 12.

Simplifying this equation gives: 7 = cos((π/6(6 - d)) + 12.

Next, solve for the phase shift (d) by finding the inverse cosine of both sides:

cos^-1(7) = (π/6(6 - d)) + 12.

Now, solve for (6 - d):

(π/6(6 - d)) = cos^-1(7) - 12.

Simplifying further: π(6 - d) = 6(cos^-1(7) - 12).

Now, solve for (6 - d):

6 - d = (6(cos^-1(7) - 12))/π.

Finally, calculate d by subtracting this value from 6:

d = 6 - (6(cos^-1(7) - 12))/π.

This value of d represents the phase shift of the cosine function as a function of time (t) in months.

dad