Find the RMS value of the function
i=15[1-e^(-1/2t)] from t=0 to t=4
A. 7.5[sqrt(1+4e^-2 -e^-4)]
B. 7.5[sqrt(4-2e^-2 +e^-4)]
C. 7.5[sqrt(4+4e^-2 -e^-4)]
D. 7.5[sqrt(5-2e^-2 +e^-4)]
If we let r = rms of f(t), then
4r^2 = Int(15 - 15e^-1/2t)^2 dt[0,4]
= 15Int(1 - e^-t/2)^2 dt[0,4]
= 15Int(1 - 2e-t/2 + e^-t)[0,4]
= 15(t + 4e^-t/2 - e^-t)[0,4]
= 15[(4 + 4/e^2 - 1/e^4) - (0+4-1)]
= 15(1 + 4/e^2 - 1/e^4)
so A.
see wikipedia on root mean square
My apologies to Alan who posted this question last week. I forgot to square f(t) in the integral.
To find the RMS value of the given function, we first need to find the square of the function, then calculate its mean value over the specified interval, and finally take the square root of the mean value.
The given function is i(t) = 15[1 - e^(-1/2t)].
1. Square the function: i^2(t) = 225[1 - 2e^(-1/2t) + e^(-t)].
2. Calculate the mean value of the squared function using integration. We need to integrate the squared function from t = 0 to t = 4 and then divide the result by the interval (4 - 0 = 4):
Mean value = (1/4) ∫(0 to 4) [225(1 - 2e^(-1/2t) + e^(-t))] dt.
3. Evaluate the integral:
Mean value = (1/4) [225(t - 2e^(-1/2t) + 2e^(-t))] evaluated from t = 0 to t = 4.
Plugging in the values, we have:
Mean value = (1/4) [225(4 - 2e^(-2) + 2e^(-4) - 0 + 0 - 2)].
Simplifying further,
Mean value = (1/4) [900 - 450e^(-2) + 450e^(-4) - 450].
4. Finally, take the square root of the mean value:
RMS value = sqrt(mean value) = sqrt((1/4) [900 - 450e^(-2) + 450e^(-4) - 450]).
Now let's simplify this expression:
RMS value = sqrt(225 - 112.5e^(-2) + 112.5e^(-4) - 112.5).
Comparing this expression with the answer choices, we can see that the correct option is:
B. 7.5[sqrt(4 - 2e^(-2) + e^(-4))].
Therefore, the RMS value of the function i(t) = 15[1 - e^(-1/2t)] from t = 0 to t = 4 is 7.5[sqrt(4 - 2e^(-2) + e^(-4))].
To find the root mean square (RMS) value of a function, you need to calculate the square root of the average of the squared values of the function over a given interval. In this case, you need to find the RMS value of the function i(t) = 15[1-e^(-1/2t)] from t=0 to t=4.
To solve this problem, follow these steps:
Step 1: Calculate the square of the function.
- Square the function i(t):
i^2(t) = [15(1-e^(-1/2t))]^2
= 225(1-2e^(-1/2t) + e^(-t))
Step 2: Integrate the squared function over the given interval.
- Integrate i^2(t) from t=0 to t=4:
∫(0 to 4) i^2(t) dt
= ∫(0 to 4) 225(1-2e^(-1/2t) + e^(-t)) dt
Step 3: Evaluate the integral.
- Evaluate the integral using appropriate techniques such as substitution or integration by parts.
Step 4: Take the square root and multiply by the appropriate constant.
- Once you have the value from the integral, take the square root of that value and multiply it by the constant factor of 15.
By following the steps above, you can determine the correct option that gives the RMS value of the function i(t).