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Over the course of a full year, the daylight in a certain city follows a periodic pattern. The graph below represents the average daylight, in
minutes, over the course of twenty-
four months, with time t, representing the number of months after January 1 of a certain year
Average Daylight (in minutes)
1000
(11.5, 951)
(23-5, 951)
900
800
700
600
300
400
(5-5, 461)
(17.5, 461)
300
200
100
10
12
14
15
Time (in months)
Write an equation in terms of y, average daylight in minutes, and t, time in months, to represent the
given context.
Answer Attempt 1 out of 2
y=
Submit Answer
B

Over The Course Of A Full Year The Daylight In A Certain City Follows A Periodic Pattern The Graph Below Represents The Average Daylight Inminutes Over The Cour class=

Sagot :

semsee45

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]

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