The normal monthly temperatures (°F) for Omaha, Nebraska, are recorded below.

Month

Jan Feb Mar

Apr May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

1

2

3

4

5

6

7

8

9

10

11

12

Temp.

21°

27°

390520620

720

770

740

650

53°

39°

25°

a. Write a sinusoidal function that models Omaha's monthly temperature variation

b. Use the model to estimate the normal temperature during the month of April.

Answer

[tex]y = 28 \sin(\frac{\pi}{6} - \frac{2\pi}{3}) + 49[/tex]

Step by Step Explanation

Given

See attachment for table

Required

Determine the sinusoidal function

The sinusoidal function is represented as:

[tex]y = A \sin(Bx + C) + D[/tex]

Where

[tex]|A| = Amplitude[/tex]

And it is calculated as:

[tex]A = \frac{1}{2}[Highest - Least][/tex]

From the table:

Highest = 77 and Least = 21

So:

[tex]A = \frac{1}{2}[77 - 21][/tex]

[tex]A = \frac{1}{2}[56][/tex]

[tex]A = 28[/tex]

B is calculated as:

[tex]Period = \frac{2\pi}{B}[/tex]

The period is 12 i.e. 12 months

So:

[tex]\frac{2\pi}{B} = 12[/tex]

[tex]B = \frac{2\pi}{12}[/tex]

[tex]B = \frac{\pi}{6}[/tex]

C is calculated as:

[tex]\frac{C}{B} = -4[/tex]

[tex]C = -4 * B[/tex]

[tex]C = -4 * \frac{\pi}{6}[/tex]

[tex]C = -2 * \frac{\pi}{3}[/tex]

[tex]C = -\frac{2\pi}{3}[/tex]

Lastly, D is calculated as:

[tex]D = Least + A[/tex]

So:

[tex]D =21 + 28[/tex]

[tex]D=49[/tex]

So, the function is:

[tex]y = A \sin(Bx + C) + D[/tex]

[tex]y = 28 \sin(\frac{\pi}{6} - \frac{2\pi}{3}) + 49[/tex]

Answer:

y=28 sin (pi/6 t - 7pi/6)+49

Step-by-step explanation:

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