Question

Household electricity in a particular country is supplied in the form of alternating current that varies from 160 V to −160 V with a frequency of 50 cycles per second (Hz). The voltage is thus given by the equation
E(t) = 160 sin(100πt)
where t is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of [E(t)]^2 over one cycle.
Calculate the RMS voltage of household current in this particular country. (Round your answer to the nearest whole number of volts.)
Many electric stoves require an RMS voltage of 220 V. Find the co

Answer 1

Find the corresponding amplitude A needed for the voltage
E(t) = A sin(100πt).

Answer 2

To find the corresponding amplitude needed for an RMS value of 220V you can just use:\[220=\frac{A}{\sqrt{2}}\]Now just solve for A :)

Answer 3

For the first part it's a little more involved. Start by knowing that the average value of a function is:\[\large Avg=\frac{1}{b-a}\int_{a}^{b}f(t)dt\]Applying that to this problem you get:\[\large Avg=\frac{1}{\frac{1}{50}}\int_{0}^{1/50}160^2sin^2(100\pi t)dt\]Ultimately you will be able to reduce this to:\[\large Avg=\frac{160^2}{2}\]That leaves you with an RMS of:\[\large RMS= \sqrt{\frac{160^2}{2}}=\frac{160}{\sqrt{2}}=113.14V\]