Let's focus on solving question number 47 in detail.
The given problem is a function that represents the cost of a product in Birr as a function of time in days: [tex]\( f(t) = 3t + t^2 \)[/tex]. The task is to find the average rate of change of this function over the interval [tex]\([2, 6]\)[/tex].
### Step-by-Step Solution:
1. Identify the interval for [tex]\( t \)[/tex]: The interval given is from [tex]\( t = 2 \)[/tex] to [tex]\( t = 6 \)[/tex].
2. Calculate [tex]\( f(t) \)[/tex] for [tex]\( t = 2 \)[/tex]:
[tex]\[
f(2) = 3(2) + 2^2 = 6 + 4 = 10
\][/tex]
3. Calculate [tex]\( f(t) \)[/tex] for [tex]\( t = 6 \)[/tex]:
[tex]\[
f(6) = 3(6) + 6^2 = 18 + 36 = 54
\][/tex]
4. Find the average rate of change: The average rate of change of the function over the interval [tex]\([2, 6]\)[/tex] can be found using the formula:
[tex]\[
\text{Average rate of change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}
\][/tex]
Here, [tex]\( t_1 = 2 \)[/tex] and [tex]\( t_2 = 6 \)[/tex].
5. Substitute the values into the formula:
[tex]\[
\text{Average rate of change} = \frac{f(6) - f(2)}{6 - 2} = \frac{54 - 10}{6 - 2} = \frac{44}{4} = 11
\][/tex]
So, the average rate of change of the cost function over the interval [tex]\([2, 6]\)[/tex] is [tex]\(\boxed{11}\)[/tex].
### Answer:
Therefore, the correct answer to this question is C. 11.
