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?

Question

asked Apr 17, 2023 206k views

1 Answer

NHG · Answer 1 · 2023-04-24T15:43:08+0000

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.

$a=\pm2$

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,

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

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.