The voltage, V, in volts, applied to an electric circuit can be modeled by the equation
V= 167sin (120 pi x)
where x is the time, in seconds. A component in the circuit can safely withstand a voltage of more than 120 V for 0.01 s or less.
Determine the length of time the voltage is greater than 120 V on each half-cycle.
set the voltage equal to 120
120 = 167sin(120pix)
.7185628 = sin(120pix)
so 120pix = .8017336 or 120pix = pi - .8017336
x = .8017336/(120pi) or x = (pi - .8017336)/(120pi)
x = .00213 or x = .00621
so it will be greater than 120 for the time period between the times of .00213 seconds and .00621 seconds
that span of time is .00621-.00213 seconds or .0041 seconds
Therefore the component is safe.
To determine the length of time the voltage is greater than 120V on each half-cycle, we need to find the values of x when V is greater than 120V.
The equation given is V = 167sin(120πx).
In the given equation, sin(120πx) will be positive when the angle (120πx) is within the range of 0 to π radians in the first half-cycle. In the second half-cycle, sin(120πx) will be negative when the angle (120πx) is within the range of π to 2π radians.
So, we need to find the values of x that satisfy the inequality 167sin(120πx) > 120.
Step 1: Solve for x in the equation sin(120πx) = 120/167.
Step 2: Determine the values of x that satisfy the inequality 167sin(120πx) > 120 for both half-cycles.
Step 1:
sin(120πx) = 120/167
120πx = arcsin(120/167)
x = arcsin(120/167) / (120π)
Note: The inverse sine function arcsin only gives a single value in the range -π/2 to π/2. To find the values of x for both half-cycles, we will find the difference between two arcsin values.
Step 2:
To find the length of time in the first half-cycle, we need to find the difference between two arcsin values:
x1 = arcsin(120/167) / (120π)
x2 = (π - arcsin(120/167)) / (120π)
length of time in first half-cycle = x2 - x1
Similarly, to find the length of time in the second half-cycle, we need to find the difference between two arcsin values:
x3 = (2π - arcsin(120/167)) / (120π)
x4 = (3π - arcsin(120/167)) / (120π)
length of time in second half-cycle = x4 - x3
Thus, the lengths of time the voltage is greater than 120V on each half-cycle are:
- First half-cycle: x2 - x1
- Second half-cycle: x4 - x3
To determine the length of time the voltage is greater than 120V on each half-cycle, we need to find the values of x for which the voltage V is greater than 120V.
The given equation is V = 167sin(120πx).
First, let's find the times when V = 120V. We can set up the equation as follows:
120 = 167sin(120πx)
Now, divide both sides of the equation by 167:
sin(120πx) = 120/167
To find the values of x, we need to take the inverse sine (or arcsin) of both sides. However, since the inverse sine function has a range of -90° to 90°, we can only find values of x between 0 and 0.5 that satisfy the equation. This is because the given equation represents one full cycle, and we only need to find the time when the voltage is greater than 120V on each half-cycle.
Let's solve for x using the inverse sine function and convert it to degrees for easier understanding:
120πx = arcsin(120/167)
x = arcsin(120/167) / (120π)
x ≈ 0.000811 seconds (rounded to 6 decimal places) or x ≈ (0.0465°)
Now that we have found the time for one half-cycle, we can find the length of time the voltage is greater than 120V in each half-cycle. Since the equation represents a sinusoidal wave, we know that within one full cycle (2π), the voltage will be greater than 120V twice, once in the positive half-cycle and once in the negative half-cycle.
Keeping this in mind, the voltage is greater than 120V for approximately 0.000811 seconds (or 0.0465°) during the positive half-cycle, and the same duration applies to the negative half-cycle.
Therefore, the length of time the voltage is greater than 120V on each half-cycle is approximately 0.000811 seconds.