We are given that the average cost $c(x)$ for printing a book is represented by the function:

$c(x) = \frac{5.5x + k}{x}$

where $x$ is the number of books printed, and $k$ is a constant. We are asked to find the value of $k$ if the average cost per book is $9 when 200 books are printed (i.e., $x = 200$).

Step-by-Step Solution:

1. Plug in the given values:
   • $c(x) = 9$ (the average cost),
   • $x = 200$ (the number of books).

$9 = \frac{5.5(200) + k}{200}$

2. Simplify the equation:

$9 = \frac{1100 + k}{200}$

3. Multiply both sides by 200 to eliminate the denominator:

$9 \times 200 = 1100 + k$

$1800 = 1100 + k$

4. Solve for $k$:

$k = 1800 - 1100$

$k = 700$

Thus, the value of $k$ is 700.

Would you like any further details on this solution?

Here are 5 related questions to expand on this topic:

1. How would the average cost change if the number of books printed was 500 instead of 200?
2. What is the minimum possible average cost as $x$ increases?
3. How would you interpret the constant $k$ in the context of this cost function?
4. If the cost per book must remain under $8, what is the minimum number of books that need to be printed?
5. How would the equation change if the variable cost per book (currently 5.5) increased by 10%?

Tip: When solving problems like these, always remember to check the units and ensure your calculations align with the context of the problem (e.g., costs per book).