Final answer:
The average rate of change of the function f(x) = ax^2 + 2ax + a over the interval [1,4] is calculated to be 7a. Setting this equal to 13 gives us a value for 'a' that does not match any of the provided options, suggesting there may be a mistake in the question.
Step-by-step explanation:
To find the value of 'a' in the function f(x) = ax^2 + 2ax + a where the average rate of change over the interval [1,4] is 13, we need to calculate the change in the function values at x=1 and x=4 and then divide by the change in x.
Firstly, find the function values at x = 1 and x = 4:
- f(1) = a(1)^2 + 2a(1) + a = a + 2a + a = 4a
- f(4) = a(4)^2 + 2a(4) + a = 16a + 8a + a = 25a
Next, calculate the average rate of change:
Average rate of change = (f(4) - f(1)) / (4 - 1) = (25a - 4a) / 3 = 21a / 3 = 7a
We are given that the average rate of change is 13, so:
7a = 13
Solve for a:
a = 13 / 7
Which simplifies to:
a ≈ 1.857, which is not any of the options provided; therefore, there might have been a calculation mistake or a typo in the options given.