3. The voltage, V, in volts, applied to an electric circuit can be modelled by the equation V=150cos(120πt) where t is time, in seconds. Determine when the voltage is 60V for the duration of one cycle. (Always consider multiple solutions)
150cos(120πt) = 60
cos(120πt) = 0.4
since cos(1.15928) = 0.4,
120πt = 1.15928 or 2π-1.15928
now finish it off
Well, the equation V = 150cos(120πt) represents a sine wave with an amplitude of 150 volts and a frequency of 60 Hz.
To determine when the voltage is 60V for the duration of one cycle, we need to find the value(s) of t when V = 60.
So, let's plug in V = 60 into the equation and solve for t:
60 = 150cos(120πt)
Dividing both sides by 150:
0.4 = cos(120πt)
Now, to find the value(s) of t, we need to find the inverse cosine of 0.4.
Using a calculator, we find that the inverse cosine of 0.4 is approximately 67.17 degrees or 1.17 radians.
However, since we are looking for the duration of one cycle, we need to consider all values of t that satisfy the equation within one complete cycle, which is 2π.
So, in order to find other solutions, we add or subtract multiples of 2π:
120πt = ±cos^(-1)(0.4) + 2πn, where n is an integer
t = (±cos^(-1)(0.4) + 2πn) / (120π), where n is an integer
So, to summarize, the voltage is 60V for the duration of one cycle at times given by:
t = (±cos^(-1)(0.4) + 2πn) / (120π), where n is an integer
Now, I hope you didn't get too shocked by this electrically charged explanation!
To determine when the voltage is 60V for the duration of one cycle, we need to find the time values where the equation V=150cos(120πt) equals 60V.
First, let's solve for t when V=60V:
60 = 150cos(120πt)
Divide both sides of the equation by 150:
0.4 = cos(120πt)
Now, we want to find the values of t that satisfy this equation for one cycle.
The cosine function has a period of 2π, meaning it repeats every 2π units. In this equation, the period is compressed by a factor of 120π, so the equivalent period is (2π)/(120π) = 1/60.
To find the values of t that correspond to V = 60V for one cycle, we can solve the equation:
0.4 = cos(2π/60 * t)
To find the solutions, we can take the inverse cosine of both sides:
cos^(-1)(0.4) = 2π/60 * t
Using a calculator, we can find the inverse cosine of 0.4 to be approximately 1.1593 radians:
1.1593 = 2π/60 * t
To solve for t, we can multiply both sides by 60/2π:
t = 1.1593 * 60 / (2π)
Simplifying this expression, we get:
t ≈ 1.1593 * 30 / π
Therefore, the time values when the voltage is 60V for the duration of one cycle are approximately t ≈ 18.5223 seconds and t ≈ 1.4787 seconds.
To determine when the voltage is 60V for the duration of one cycle, we need to find the values of t that satisfy the equation V = 60.
Let's start by substituting V = 60 into the equation V = 150cos(120πt):
60 = 150cos(120πt)
Next, divide both sides of the equation by 150:
0.4 = cos(120πt)
Now, we need to find the values of t that satisfy the equation cos(120πt) = 0.4.
To do this, we can take the inverse cosine (cos⁻¹) of both sides of the equation:
cos⁻¹(0.4) = 120πt
Using a calculator, we find that the inverse cosine of 0.4 is approximately 1.1593 radians.
So, we have:
1.1593 = 120πt
To isolate t, divide both sides of the equation by 120π:
t = 1.1593 / (120π)
Now, to determine the duration of one cycle, we need to find the time it takes for the cosine function to complete a full cycle. A cosine function has a period of 2π, which means it completes one full cycle within 2π units of time.
Therefore, the duration of one cycle is 2π.
To find other values of t that satisfy the equation V = 60V for the duration of one cycle, we can add or subtract multiples of the duration of one cycle. So, the general solution for t is given by:
t = (1.1593 / (120π)) + n(2π)
where n is an integer representing the number of cycles.
To find specific solutions, you can substitute different values for n, such as 0, ±1, ±2, etc., into the equation. This will give you different values of t at which the voltage is 60V for the duration of one cycle.