The image you've uploaded contains a math problem related to marginal cost. The problem is asking you to find the cost of producing x items given a cost function, where the marginal cost is $1.75 - 0.002x$ and the cost of producing 1 item is $562.

Here’s the solution approach:

1. Understanding Marginal Cost: The given equation describes the marginal cost of producing x items. Marginal cost is the rate of change of the total cost with respect to the number of items produced. To find the total cost, we need to integrate the marginal cost function.

2. Set up the equation:
   The marginal cost function is $\frac{dC}{dx} = 1.75 - 0.002x$.
   To find the total cost $C(x)$, integrate the marginal cost function: $C(x) = \int (1.75 - 0.002x) dx$

3. Integrate: $C(x) = 1.75x - 0.001x^2 + C_0$ Here, $C_0$ is a constant of integration, which we can find using the given condition that the cost of producing 1 item is $562.

4. Use the given condition:
   We know that $C(1) = 562$, so substitute $x = 1$ into the cost function to solve for $C_0$: $562 = 1.75(1) - 0.001(1)^2 + C_0$ $562 = 1.75 - 0.001 + C_0$ $C_0 = 562 - 1.749 = 560.251$

5. Final cost function: Now that we have $C_0$, the cost function is: $C(x) = 1.75x - 0.001x^2 + 560.251$

6. Find the cost of producing 100 items: Substitute $x = 100$ into the cost function: $C(100) = 1.75(100) - 0.001(100)^2 + 560.251$ $C(100) = 175 - 0.001(10000) + 560.251$ $C(100) = 175 - 10 + 560.251 = 725.251$

So, the cost of producing 100 items is approximately $725.25.

Would you like any further explanation or clarifications about this problem?

Here are 5 related questions to explore further:

1. How does the marginal cost function relate to the total cost function?
2. What does the constant $C_0$ represent in this context?
3. How would you find the average cost per item for producing 100 items?
4. What happens to the cost if the marginal cost function changes (e.g., different coefficients)?
5. Can you apply the same method to a cost function with more complex terms?

Tip: When solving problems with marginal cost, remember that the constant of integration represents the fixed cost when no items are produced.