Question

common household electric current is called alternating current because the current alternates direction within the wires. the voltage v in a typical 115 - volt outlet can be expressed by the function v(t)=163 sin ωt, where ω is the angular speed (in radians per second) of the rotating generator at the electrical plant and t is the time measured in seconds. it is essential for electric generators to rotate at precisely 60 cycles per sec. complete parts (a) and (b). (a) how many times does the current oscillate in 0.15 sec? □ time(s) (b) what are the maximum and minimum voltages in this outlet? is the voltage always equal to 115 volts? the maximum voltage is □ v and the minimum voltage is □ v. is the voltage always equal to 115 volts? no yes

Explanation:

Step1: Find the number of cycles in 0.15 sec

The generator rotates at 60 cycles per second.
To find the number of cycles in \(t = 0.15\) seconds, we use the formula \(n=\text{cycles per second}\times t\).
So \(n = 60\times0.15\).

Step2: Calculate the value of \(n\)

\(n=60\times0.15 = 9\)

Step3: Find the maximum and minimum of \(y = A\sin(\omega t)\)

For the function \(V(t)=163\sin(\omega t)\), the general form of a sine function is \(y = A\sin(Bx - C)+D\). Here \(A = 163\), \(B=\omega\), \(C = 0\), \(D = 0\).
The range of the sine function \(y=\sin(x)\) is \([- 1,1]\).
When \(\sin(\omega t)=1\), \(V(t)_{\text{max}}=163\times1 = 163\)
When \(\sin(\omega t)=-1\), \(V(t)_{\text{min}}=163\times(-1)=-163\)

Answer:

(a) \(9\) time(s)
(b) The maximum voltage is \(163\) V and the minimum voltage is \(- 163\) V.
No (for the question "Is the voltage always equal to 115 volts?")