an electric current I , in amps is given by:
I= cos(wt) + sqrt of 3 sin(wt)
where w is positive constant. what are the max and minimum values of I?
*please be specific
dI/dt = -wsin wt + w√3cos wt
= 0 for max min
wsin wt = w√3cos wt
sin wt/cos wt = w√3/w = √3
tan wt = pi/3 or 4pi/3
if wt = pi/3
I = cos pi/3 + √3sin pi/3 = .5 + 1.5 = 2
if wt = 4pi/3
I = cos 4pi/3 + √3sin 4pi/3 = -.5 -1.5 = -2
The pi/3 yields the maximum, notice I = +2
and the 4pi/3 yields the minimum
The actual question was, "what are the max and min values of I"
max of I is 2
min of I is -2
the pi/3 and 4pi/3 produce those max and mins, they themselves are not the max and mins.
so it could be pi/3 or 4pi/3
which means both are correct?
should i put both answers or one answer on my final paper?
Well, this electric current equation seems to have a sinusoidal nature. To find the maximum and minimum values, we need to examine the amplitude of each component separately.
The general form of the equation is I = A*cos(wt) + B*sin(wt), where A and B are the amplitudes of the cosine and sine components, respectively.
In this case, A = 1 and B = √3. To find the maximum and minimum values, we can calculate the amplitude of the resultant current using the Pythagorean theorem.
The amplitude (Amp) of the current is given by:
Amp = sqrt(A^2 + B^2)
Plugging in the values we have:
Amp = sqrt(1^2 + (√3)^2)
Amp = sqrt(1 + 3)
Amp = sqrt(4)
Amp = 2
So, the maximum value of I is 2 amps (when the cosine and sine components both reach their maximum values), and the minimum value of I is -2 amps (when the cosine and sine components both reach their minimum values).
To find the maximum and minimum values of the given current function, we need to analyze the coefficients of the cosine and sine terms separately.
The given current function is:
I = cos(wt) + √3 sin(wt)
We can rewrite this current equation in the form of A cos(ϕ - α), where A is the maximum amplitude and α is the phase angle.
To do that, let's find the equivalent amplitude A first:
A = √(coefficient of cos^2(wt) + coefficient of sin^2(wt))
= √(1^2 + (√3)^2)
= √(1 + 3)
= √4
= 2
So, the maximum amplitude (A) of the given current function is 2.
Next, let's find the phase angle (α):
tan(α) = (coefficient of sin(wt)) / (coefficient of cos(wt))
= √3 / 1
= √3
Since √3 is positive and tan(α) is positive for α in the first and third quadrants, we can say that α = π/3.
Therefore, the current function can be rewritten as:
I = 2 cos(wt - π/3)
Now, we can determine the maximum and minimum values of I by considering the amplitude (2) and the phase angle (π/3).
The maximum value of I occurs at wt - π/3 = 0, which means wt = π/3 (since cos(0) = 1):
I_max = 2 cos(π/3)
= 2 * (1/2)
= 1
So, the maximum value of I is 1 Amp.
The minimum value of I occurs at wt - π/3 = π, which means wt = 4π/3 (since cos(π) = -1):
I_min = 2 cos(4π/3)
= 2 * (-1/2)
= -1
Therefore, the minimum value of I is -1 Amp.
In summary, the maximum value of the given current function is 1 Amp, and the minimum value is -1 Amp.