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 0.25 sec?
15 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 163 v and the minimum voltage is - 163 v.
is the voltage always equal to 115 volts?
no
yes

Explanation:

Step1: Find the number of cycles

The formula for the number of cycles \(n\) in time \(t\) is \(n = ft\), where \(f\) is the frequency. Given \(f = 60\) cycles per second and \(t=0.25\) s.
\(n=60\times0.25\)

Step2: Calculate the value of \(n\)

\(n = 15\)

Step3: Analyze the voltage function \(V(t)=163\sin(\omega t)\)

The sine function \(y = A\sin(Bx - C)+D\) has a maximum value of \(A + D\) and a minimum value of \(-A+D\). Here \(A = 163\), \(D = 0\). The maximum value of \(V(t)\) is \(163\) and the minimum value is \(- 163\). The voltage \(V(t)\) is a periodic function. The value \(V(t)=115\) occurs when \(163\sin(\omega t)=115\), i.e., \(\sin(\omega t)=\frac{115}{163}\). Since the sine function \(y = \sin x\) is not a constant function (\(y=\sin x\in[- 1,1]\) and it oscillates), the voltage is not always \(115\) volts.

Answer:

(a) \(15\)
(b) The maximum voltage is \(163\) V and the minimum voltage is \(-163\) V. No, the voltage is not always equal to \(115\) volts.