What is the average value of the voltage induced in a tape head over the interval from t = 0 to t = 3?
To find the average value of the voltage v over the interval from t = 0 to t = 3, we need to evaluate the definite integral of the function v = t^2 e^3t with respect to t over this interval. The average value is given by the formula:
Average value = (1 / (b - a)) * ∫(a to b) v dt
where a and b represent the starting and ending points of the interval, respectively. Substituting the given values a = 0 and b = 3 into the formula, we have:
Average value = (1 / (3 - 0)) * ∫(0 to 3) t^2 e^3t dt.
To evaluate this integral, we can use integration techniques such as integration by parts or tables of integrals. Once the integration is performed, we round the result to the nearest volt. In this case, the average value of v is approximately 6502 volts, which corresponds to option C).
Understanding the Calculation Process
When determining the average value of the voltage induced in a tape head over a specific time interval, we utilize the concept of definite integrals to compute the average. In this particular scenario, the voltage function is given by v = t^2 e^3t, where t represents the time in seconds.
By utilizing the average value formula mentioned earlier, we calculate the integral of the voltage function over the interval from t = 0 to t = 3. This involves evaluating the definite integral of t^2 e^3t with respect to t. Integration techniques such as integration by parts or tables of integrals are typically employed to solve the integral.
Once the integration is completed, we round the result to the nearest volt to obtain the average value of the voltage over the specified time interval. In this case, the average value of the voltage induced in a tape head from t = 0 to t = 3 is approximately 6502 volts.
Understanding the process of calculating the average value of a function over a specific interval is essential in various mathematical applications, including electrical engineering and signal processing. It allows us to determine meaningful insights and make informed decisions based on the given data.
In conclusion, the average value of the voltage induced in a tape head over the interval from t = 0 to t = 3 can be accurately calculated using definite integrals and appropriate mathematical techniques. This calculation yields valuable information about the system's performance and behavior during the specified time period.