hi can anyone can help me on these 2 questions

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     

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

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

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