A sound wave is given by the following equation:
y = 6 sin(324pi t)
where t = time in seconds
How many cycles will occur
between t= 3 and t = 5.5 seconds?

405 cycles

Step-by-step explanation:

We have the equation:

y = 6*sin(324*π*t)

For the properties of the sin function, we know that the period is 2π.

So between:

Sin(x) and Sin(x + 2*pi)

we have a cycle.

between:

Sin(x) and Sin(x + 6*pi)

we have 3 cycles.

and so on.

Now we want to find how many cycles will occur between t = 3 s, and t = 5.5 seconds

Between these times, the difference in the argument of the sin function is:

324*π*5.5 - 324*π*3 = 324*π*(5.5 - 3) = 324*π*2.5

Now, the number of cycles that we will have between these times is equal to the number of times that "2*π" is in 324*π*2.5

That number is just the quotient between 324*π*2.5 and 2*π

N = (324*π*2.5)/(2*π) = (324*2.5)/(2) = 405

There are 405 cycles between 3 seconds and 5.5 seconds.