To solve the problem, we use the given cost function:

$C(x) = 7.25x + 2500$

where:

• $C(x)$ represents the total daily cost (in dollars),
• $x$ represents the number of items manufactured.

Part a: Finding $C(0)$

We substitute $x = 0$ into the function to find $C(0)$:

$C(0) = 7.25(0) + 2500 = 0 + 2500 = 2500$

Explanation of the result

The result $C(0) = 2500$ means that when no items are manufactured (i.e., $x = 0$), the total daily cost is $2500. This value represents the fixed cost that the company incurs daily, regardless of how many items are produced. These costs might include expenses like rent, salaries, utilities, and other overheads that do not depend on production levels.

Would you like further details or have any other questions? Here are some related questions for practice:

1. What would the total cost be if 100 items are manufactured?
2. What is the slope of the function $C(x)$, and what does it represent?
3. How much does the cost increase for each additional item produced?
4. What would the total cost be if 500 items are manufactured?
5. How could this function be used to predict future costs for different production levels?

Tip: In linear cost functions like this, the fixed cost is always the constant term (in this case, 2500), and the variable cost per item is the coefficient of $x$.