Final answer:
The average value of the function f(x) = 25 - x² on the interval [0, 3] is calculated by integrating the function over the interval and dividing by the length of the interval. The result obtained is 22, but this option is not provided in the given choices, implying a possible error in the options or the calculations.
Step-by-step explanation:
The question asks us to find the average value of the function f(x) = 25 - x² on the interval [0, 3]. To find the average value of a function over a given interval, we integrate the function over the interval and then divide by the length of the interval.
The integral of f(x) from 0 to 3 is:
∫(25 - x²)dx = [25x - (x³/3)] |_0^3 = [25(3) - (3³/3)] - [25(0) - (0³/3)] = 75 - 9 = 66
Now, we find the average value by dividing the integral by the length of the interval, which is 3 - 0 = 3.
Average value = ∫(25 - x²)dx / (3 - 0) = 66 / 3 = 22
However, the result obtained (22) is not present in the given options. Thus, either there is a mistake in my calculations or the given options are incorrect. Therefore, I cannot select any of the provided options (a. 73.33, b. 25, c. 12.5, d. 0) as the correct average value for the function over the interval.