the voltage, v, in volts, applied to an electric circuit can be modelled by the equation v=167sin(120pi)(t), where t is the time, in seconds. A component in the circuit can safely withstand a voltage of more than 120 V for 0.01 sec or less.
a) determine the length of the time that the voltage is greater than 120 V on each half-cycle.
b)is it safe to use this component in this circuit? justify your answer.
can you please explain to me clearly how I to do this q
To determine the length of time that the voltage is greater than 120V on each half-cycle, we need to find when the voltage function v(t) exceeds 120V.
The given equation for the voltage is v = 167sin(120πt), where t is the time in seconds.
a) To find the length of time, we need to solve the inequality:
167sin(120πt) > 120
First, we will divide both sides by 167 to simplify the inequality:
sin(120πt) > 120/167
Now, we need to isolate t. To do that, we will take the inverse sine (or arcsine) of both sides:
arcsin(sin(120πt)) > arcsin(120/167)
120πt > arcsin(120/167)
To solve for t, divide both sides by 120π:
t > (1/(120π)) * arcsin(120/167)
Now we can calculate the value of t using this expression. Note that we are only interested in the positive values of t since it represents time:
t > (1/(120π)) * arcsin(120/167)
Using a calculator, we find that t ≈ 0.003782 seconds.
Since we are looking for the length of time for each half-cycle, we multiply this value by 2:
Length of time = 2 * t ≈ 2 * 0.003782 seconds ≈ 0.0076 seconds
Therefore, the voltage is greater than 120V for approximately 0.0076 seconds on each half-cycle.
b) To determine if it is safe to use this component in the circuit, we compare the calculated length of time with the given threshold of 0.01 seconds.
The calculated time (0.0076 seconds) is less than the given threshold (0.01 seconds). Therefore, the component can safely withstand the voltage in this circuit.
In summary, the length of time that the voltage exceeds 120V on each half-cycle is approximately 0.0076 seconds, and it is safe to use the component in this circuit.