Question

What is the smallest time t such that I = 4?

A generator produces an alternating current according to the equation I = 8 sin 124 `pi` t, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 4?

Answer 1

`t~~0.0013 seconds`

Explanation:

`4=8sin 124pi t`

`4/8 =sin 124 pi t`

`sin^-1 (1/2) =124 pi t`

`124 pi t = pi/6 + 2pin, or 124 pi t = (5pi)/6 + 2pin`

`t=(pi/6 + 2pin)/(124pi) or t =((5pi)/6 + 2pin)/(124 pi)`

`t=(pi/6 + 2pin) *1/(124pi) or t=((5pi)/6 + 2pin)*1/(124 pi)`

`t=1/744 +1/62 n or t=5/744 +1/62 n` where `n=0,+-1,+-2,+-3,...`

Since time is positive we are looking for the first positive answer. So pick n values and plug it to the two equations.

`n =0, t~~0.0013 or t~~.00672`

Note that if we pick n = -1 then we get two negative answers and if we pick n = 1 then we get 0.0175 and 0.02285 which are bigger than the values for n = 0 so the smallest time t when I = 4 is about 0.0013 sec.