Question

A message in a bottle is floating on top of the ocean in a periodic manner. The time between periods of maximum heights is 24 seconds, and the average height of the bottle is 8 feet. The bottle moves in a manner such that the distance from the highest and lowest point is 4 feet. A cosine function can model the movement of the message in a bottle in relation to the height. 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?

Answer 1

We are given that the movement of a bottle can be modeled using a cosine function. The general form of a cosine function is the following:

`y=A\cos (k(t-b))+c`

Where:

`\begin{gathered} A=\text{ amplitude} \\ k=\text{ }periodicity \\ c=\text{ }vertical\text{ shift} \\ b=\text{ horizontal shift} \end{gathered}`

First, we will determine the periodicity "k". To do that we need to have into account that the time between height points is 24 seconds. This is known as the period and is related to the periodicity through the following formula:

`T=(2\pi)/(k)`

Therefore, if we solve for "k" by multiplying by "k" on both sides:

`kT=2\pi`

And dividing both sides by "T":

`k=(2\pi)/(T)`

We get a formula for "k". Substituting we get:

`k=(2\pi)/(24)`

Simplifying we get:

`k=(\pi)/(12)`

Substituting in the cosine wave we have:

`y=A\cos ((\pi)/(12)(t-b))+c`

Now, the value of "c" is the average height of the bottle since "c" is the distance from the height 0 to the middle of the wave, therefore, we have:

`y=A\cos ((\pi)/(12)(t-b))+8`

Now, we determine the amplitude. The amplitude is the distance from the middle of the cosine wave to its highest point. Therefore, we need to average the distance between the highest and lowest points:

`A=(4ft)/(2)=2ft`

Substituting in the cosine function:

`y=2\cos ((\pi)/(12)(t-b))+8`

Now, to determine the value of "b we use the fact that at "t = 0" the height is "y = 8". Substituting we get:

`8=2\cos ((\pi)/(12)(0-b))+8`

Now, we subtract 8 from both sides:

`0=2\cos ((\pi)/(12)(0-b))`

We divide both sides by 2:

`0=\cos ((\pi)/(12)(-b))`

Now, we take the inverse function of cosine:

`\cos ^(-1)(0)=(\pi)/(12)(-b)`

Solving the operations:

`(\pi)/(2)=(\pi)/(12)(-b)`

Now, we cancel out the pi:

`(1)/(2)=(1)/(12)(-b)`

Multiplying both sides by 12:

`\begin{gathered} (12)/(2)=-b \\ 6=-b \end{gathered}`

Multiplying both sides by -1:

`-6=b`

Substituting we get:

`y=2\cos ((\pi)/(12)(t+6))+8`

We are also given that at the beginning the function moves upwards, therefore, we need to use the negative horizontal shift to fulfill this condition, therefore, we have:

`y=2\cos ((\pi)/(12)(t-6))+8`

And thus we get the equation. The graph looks like this: