Answer:
[tex]y=245\cos\left(\dfrac{\pi}{6}(x-11.5)\right)+706[/tex]
Step-by-step explanation:
To write an equation representing the given scenario, we can use the general form of the cosine function:
[tex]y=A\cos(B(x+C))+D[/tex]
where:
- |A| is the amplitude (distance between the midline and the peak).
- 2π/B is the period (horizontal distance between consecutive peaks).
- C is the phase shift (horizontal shift where negative is to the right).
- D is the vertical shift (y = D is the midline).
The peaks and troughs of the curve are:
- Peaks: (11.5, 951) and (23.5, 951)
- Troughs: (5.5, 461) and (17.5, 461)
The amplitude (A) is half the distance between the maximum and minimum values. Given that the maximum value is 951 and the minimum value is 461, then:
[tex]A=\dfrac{951-461}{2}=\dfrac{490}{2}=245[/tex]
The vertical shift (D) is the average of the maximum and minimum values. Therefore:
[tex]D=\dfrac{951+461}{2}=\dfrac{1412}{2}=706[/tex]
The period (2π/B) is the distance between two consecutive peaks (or troughs). To determine the period, subtract the x-coordinate of the first peak (11.5) from the x-coordinate of the second peak (23.5):
[tex]\dfrac{2\pi}{B}=23.5-11.5 =12[/tex]
Now, solve for B:
[tex]\dfrac{2\pi}{B}=12 \\\\\\ B=\dfrac{2\pi}{12} \\\\\\ B=\dfrac{\pi}{6}[/tex]
The phase shift (C) is the horizontal shift from the standard position.
A peak of the parent cosine function y = cos(x) occurs at x = 0. Since the first peak of the graphed function is at t = 11.5, the function has been shifted 11.5 units to the right. Therefore:
[tex]C=-11.5[/tex]
Substitute the values of A, B, C and D into the general equation of the cosine function:
[tex]y=245\cos\left(\dfrac{\pi}{6}(x-11.5)\right)+706[/tex]
So, the equation that represents the average daylight in minutes (y) is:
[tex]\Large\boxed{\boxed{y=245\cos\left(\dfrac{\pi}{6}(x-11.5)\right)+706}}[/tex]
