The current, I, in amperes, for an electric circuit is given by the formula I=4.3sin120(pi)t, where t is time, in seconds.
a)At what time is the current at its max volume? How does your understanding of co terminal angles help in your solution?
b)What time is the current at its minimum value?
this makes no sense to me, and I have no idea how I can incorporate trig into this. Please give a thorough explanation! Thanks.
I has a period of 1/60 seconds. Since it starts at zero, it is a maximum 1/4 period into each cycle, or t=1/240. Then again every 1/60 of a second.
It is a minimum at half a period, or 1/120 seconds after each maximum.
Just review the properties of the sine function.
Well, don't worry, I'm here to help you with a clownishly good explanation!
a) To find the time when the current is at its maximum volume, we need to find the value of t that makes the sine function reach its maximum, which is 1. But before we do that, let's talk about coterminal angles!
Coterminal angles are angles that have the same initial and terminal sides but differ by a multiple of 360 degrees (or 2π radians). For example, 120 degrees and 480 degrees are coterminal angles because you can get from one to the other by rotating in a full circle.
Now, back to the problem. Since the sine function has a period of 2π, we need to find the angle (120π)t that, when multiplied by 120 and π, makes the sine function reach its maximum of 1. We know that sin(π/2) = 1, so we want (120π)t to be equal to π/2.
Let's solve this equation:
(120π)t = π/2
Divide both sides by 120π:
t = (π/2) / (120π)
Cancel out the π:
t = 1 / (2*120)
Calculate:
t = 1 / 240
So, the time when the current is at its maximum volume is t = 1/240 seconds.
b) To find the time when the current is at its minimum value, we need to find the value of t that makes the sine function reach its minimum, which is -1. Using the same logic as before, we want (120π)t to be equal to -1.
Let's solve this equation:
(120π)t = -1
Divide both sides by 120π:
t = -1 / (120π)
Calculate:
t = -1 / 120π
So, the time when the current is at its minimum value is t = -1/(120π) seconds.
Remember, time can be negative or positive, but I'm afraid negative time machines don't exist yet. So, keep in mind that negative time values might not be realistic in this context.
Hope this explanation helped, and remember, always circulate your funny bone!
To solve this problem, we need to understand the behavior of the sine function and its relationship with angles and time. Let's break it down step-by-step:
a) To find the time when the current is at its maximum value, we need to find the value of "t" that maximizes the sine function.
The sine function repeats its values every 2π radians (or 360 degrees). In this case, the coefficient of "t" inside the sine function is 120π.
Since one complete cycle of the sine function occurs every 2π radians, we can find the time for a complete cycle by dividing 2π by 120π:
Time for one complete cycle = 2π / 120π = 1 / 60
This means that the current completes one full cycle every 1/60 seconds.
Now, to find the time when the current is at its maximum value, we need to divide the period (1/60) by 4 to find the quarter period. This is because the sine function reaches its maximum value (1) at quarter intervals within a complete cycle.
Quarter Period = (1/60) / 4 = 1 / (60 * 4) = 1 / 240
Therefore, the time when the current is at its maximum value is (1/240) seconds.
Understanding co-terminal angles is not relevant in this case since we are not dealing with angles directly. The "t" in the equation represents time, not angles.
b) Similarly, to find the time when the current is at its minimum value, we need to find the value of "t" that minimizes the sine function, which occurs at the bottom of each cycle.
The sine function reaches its minimum value (-1) at three-quarter intervals within a complete cycle. Therefore, we need to find the time for a three-quarter period.
Three-Quarter Period = (3/4) * (1/60) = 3/(4*60) = 1 / (80)
Therefore, the time when the current is at its minimum value is (1/80) seconds.
In summary:
a) The current is at its maximum value at (1/240) seconds.
b) The current is at its minimum value at (1/80) seconds.
To find the time at which the current is at its maximum or minimum value, we need to understand the concept of trigonometric functions and how they relate to periodic phenomena.
In this case, the equation for current (I) in the electric circuit is given by I = 4.3sin(120πt), where t is the time in seconds.
a) To find the time at which the current is at its maximum value, we need to determine when the sine function reaches its maximum value of +1. The sine function takes on its maximum value of +1 at 90 degrees and multiples of 360 degrees.
However, since we are working with radians, we need to convert these angles to radians. One full revolution in radians is equal to 2π radians. So, to find the time at which the current is at its maximum value, we need to solve the equation:
4.3sin(120πt) = 1
Divide both sides of the equation by 4.3:
sin(120πt) = 1/4.3
Now, take the inverse sine (arcsin) of both sides to find the angle:
120πt = arcsin(1/4.3)
Next, divide both sides by 120π to isolate t:
t = arcsin(1/4.3) / (120π)
Using a calculator, evaluate the right-hand side to find the time at which the current is at its maximum volume.
As for the understanding of coterminal angles, they are angles that have the same initial and terminal sides but differ by a multiple of 360 degrees or 2π radians. In this case, since we are solving for t in radians, knowing about coterminal angles doesn't significantly help in finding the time at which the current is at its maximum volume.
b) Similarly, to find the time at which the current is at its minimum value, we need to determine when the sine function reaches its minimum value of -1. The sine function takes on its minimum value of -1 at 270 degrees and multiples of 360 degrees.
Following the same steps as above, we solve the equation:
4.3sin(120πt) = -1
Divide both sides by 4.3:
sin(120πt) = -1/4.3
Take the inverse sine (arcsin) of both sides:
120πt = arcsin(-1/4.3)
Divide both sides by 120π to isolate t:
t = arcsin(-1/4.3) / (120π)
Evaluate the right-hand side using a calculator to find the time at which the current is at its minimum value.
In summary, to find the times at which the current is at its maximum or minimum values, we used the equation for current and solved for the respective values of the sine function. Understanding the concept of coterminal angles is not crucial in this particular problem.