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Assuming that at t = 0 the message in a bottle is at its average height and moves upwards after, what is the equation of the function that could represent the situation?

Assuming that at t = 0 the message in a bottle is at its average height and moves-example-1
User Genry
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1 Answer

5 votes

Given that

The period is 24 seconds

The average height is 8 feet.

The distance between the maximum and minimum height is 4 feet.

Cosine function for this model.

At t=0 the function is in average height.

Let y be the height.

Let t be the time of seconds

The general cosine equation is


y=a\cos (bt)+c

Here amplitude |a| represents the half distance between the maximum and minimum height.


|a|=(4)/(2)=2


a=\pm2

Substitute a=2 in the general equation, we get


y=2\cos (bt)+c
\text{Period =}(2\pi)/(|b|)

Substitute period =24, we get


\text{24 =}(2\pi)/(|b|)

Using the cross product, we get


24|b|=2\pi

Dividing both sides by 24, we get


|b|=(2\pi)/(24)=(\pi)/(12)
b=\pm(\pi)/(12)

Substitute b=pi/12 in the general equation, we get


y=2\cos ((\pi t)/(12))+c

When t=0 the height is 8,


8=2\cos ((\pi(0))/(12))+c


8=2(1)+c
c=8-2=6
c=6

Substitute c=6 in the equation, we get


y=2\cos ((\pi t)/(12))+6

We get the equation for the average height, it will increase upward and reach the highest height.

User NHG
by
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