Question

How do you find the amplitude for `y(t) = 1/4e^-t cos6t`?

Answer 1

In this damped oscillation, as in springs, the amplitude
`(1/4 e^(- t )) darr` as `t uarr`. See graphs for both the oscillation and the diminishing amplitude.

Explanation:

In the amplitude-damping periodic oscillation,

`y = (1/4e^(-t)) cos 6t, t >= 0`,

t-intercepts;`( 2k + 1 ) pi/2` ( from zeros of `cos 6t` ,

`k = 0, 1, 2, 3, ...`

the amplitude `a ( t ) = 1/4e^(-t)`

is a function of time t that `darr`, as `t uarr`.

The period is `(2pi)/6 = pi/3`.

See the second graph, for the exponential decay of amplitude..
graph{y-1/4e^(-x) cos (8x) = 0[0 6 -1.5 1.5]}
Amplitude-decay graph:
graph{(y-1/4e^(-x))(y-0.04) = 0[0 5 0 1]}

See just one-in-many referenceas:

https://deutsch.physics.ucsc.edu/6A/book/harmonic/node18.html