Question

An object oscillates 4 feet from its minimum height to its maximum height. The object is back at the maximum height every 3 seconds. Which cosine function may be used to model the height of the object?

Answer 1

`y=2\text{cos}(((2\pi)/(3))t)`

Explanation:

We have been given that an object oscillates 4 feet from its minimum height to its maximum height. The object is back at the maximum height every 3 seconds. We are asked to find the cosine function that can be used to model the height of the object.

We know that standard form of cosine function is
`y = A\cdot \text{cos}(Bt-C)+D`, where,

|A| = Amplitude,

Period =
`(2\pi)/(|B|)`,

C = Phase shift,

D = Vertical shift.

Since distance between maximum and minimum is 4, therefore, amplitude will be half of it, that is,
`A = 2`.

Since objects gets back to its maximum value in every 3 seconds, therefore, period of the function is 3 seconds. We know that period is given by
`(2\pi)/(|B|)`, therefore, we can write
`(2\pi)/(|B|)=3`, therefore,
`B = (2\pi)/(3)`.

We haven't been given any information about phase and mid-line, we can assume the values of C and D to be zero .

Therefore, our function required function would be
`y=2\text{cos}(((2\pi)/(3))t)`.