For the cost function, find the marginal cost at the given production level x. State the units of measurement. (All costs are in dollars.) HINT [See Example 1.]


c(x) = 15000 +90x +1000/x   x = 100

To find the marginal cost at the given production level \(x = 100\) for the cost function \(c(x) = 15000 + 90x + \frac{1000}{x}\), we need to calculate the derivative of the cost function with respect to \(x\), which represents the marginal cost function \(c'(x)\). Then, we will substitute \(x = 100\) into \(c'(x)\) to find the marginal cost at that production level.

### Step 1: Differentiate the cost function
Given \(c(x) = 15000 + 90x + \frac{1000}{x}\), let's calculate the derivative \(c'(x)\).

\[
c'(x) = \frac{d}{dx} \left(15000 + 90x + \frac{1000}{x}\right)
\]

The derivative of each term is as follows:
- The derivative of \(15000\) is \(0\) (since it's a constant).
- The derivative of \(90x\) is \(90\).
- The derivative of \(\frac{1000}{x}\) is \(-\frac{1000}{x^2}\).

So, the marginal cost function is:

\[
c'(x) = 90 - \frac{1000}{x^2}
\]

### Step 2: Calculate the marginal cost at \(x = 100\)

Now, substitute \(x = 100\) into the marginal cost function:

\[
c'(100) = 90 - \frac{1000}{(100)^2} = 90 - \frac{1000}{10000} = 90 - 0.1 = 89.9
\]

### Final Answer:
The marginal cost at \(x = 100\) is **89.9 dollars per unit**.

Would you like more details on how the derivative was calculated?

### Related Questions:
1. How is the marginal cost interpreted in real-world scenarios?
2. What happens to the marginal cost as production levels increase?
3. How does the term \(\frac{1000}{x}\) influence the cost function?
4. Can you explore the behavior of marginal cost as \(x \to 0\)?
5. How would the marginal cost change if we modified the constants in the cost function?

### Tip:
Marginal cost represents the cost of producing one additional unit at a given production level. It helps businesses determine the optimal production quantity.

For the cost function, find the marginal cost at the given production level x. State the units of measurement. (All costs are in dollars.) HINT [See Example 1.] c(x) = 15000 +90x +1000/x   x = 100

Learn how to calculate the marginal cost for the cost function c(x) = 15000 + 90x + 1000/x at a production level of x = 100. This detailed guide explains how to differentiate the cost function and interpret the marginal cost, suitable for undergraduate students in economics or calculus.

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