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1. A leaf floats on the water bobbing up and down. The distance between its highest and lowest point is 4 centimeters. It moves from its highest point down to its lowest point and back to its highest point every 10 seconds. Write a cosine function that models the movement of the leaf in relationship to the equilibrium point.

**I got y = 4 cos(π/5)m as my answer - where y is the movement of the leaf and m is the relationship of the leaf to the equilibrium point, but I'm not sure if it is right. 

 

2. The mean average temperature in Buffalo NY is 47.5 degrees. The temperature fluctuates 23.5 degrees above and below the mean temperature. If t =1 represents January, the phase shift of the sine function is 4. 

a. write a model for the average monthly temperature in Buffalo.

b. according to the model, what is the average temperature in March?

c. according to the model, what is the average temperature in August?

**I didn't know how to solve this one...if anyone can help me on these two questions, thank you so much!!!!

Guest Mar 26, 2018
 #1
avatar+94100 
+2

1. A leaf floats on the water bobbing up and down. The distance between its highest and lowest point is 4 centimeters. It moves from its highest point down to its lowest point and back to its highest point every 10 seconds. Write a cosine function that models the movement of the leaf in relationship to the equilibrium point.

**I got y = 4 cos(π/5)m as my answer - where y is the movement of the leaf and m is the relationship of the leaf to the equilibrium point, but I'm not sure if it is right. 

 

\(y=acos(nt)\)

 

a is the ampitude and that is 2

It is 2 cm from the centre of the wave to the crest (or trough)

 

The angle is in radians. if n=1 the period is 2pi   

in general though the period is 2pi/n

 

The question requires that the period is 20     

\(20=\frac{2\pi}{n}\\ 20n=2\pi\\ 10n=\pi\\ n=\frac{\pi}{10}\)

 

\(y=2cos(\frac{\pi t}{10})\qquad \text{where t is in seconds}\\\)

Melody  Mar 26, 2018
 #2
avatar+94100 
+2

2.

The mean average temperature in Buffalo NY is 47.5 degrees. The temperature fluctuates 23.5 degrees above and below the mean temperature. If t =1 represents January, the phase shift of the sine function is 4. 

a. write a model for the average monthly temperature in Buffalo.

 

let x=month where December is 0 and 12 etc

let y= temperature

 

let a= amplitude

let h=average temperature

The wavelength will be 2pi/n

 

\(y=asin[n(x-k)]+h \)

 

h=47.5

a=23.5

k=4

 

The wavelength is 12 months and wavelength = 2pi/n

so

\(12=\frac{2\pi}{n}\\ n=\frac{2\pi}{12}=\frac{\pi}{6}\)

 

So we have:

\(y=asin[n(x-k)]+h\\ y=23.5sin[\frac{\pi}{6}(x-4)]+47.5\\ \)

 

b. according to the model, what is the average temperature in March?    x=3

23.5*sin(pi/6(3-4))+47.5 = 35.75

So, according to the modal the average March temp is 36 degrees

 

c. according to the model, what is the average temperature in August?     x=8

23.5*sin(pi/6(8-4))+47.5 = 67.851596988924

So, according to the modal the average August temp is   68 degrees

 

Here is the graph

Melody  Mar 26, 2018

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