**Round your answers to two decimal places.**
**** 
**Suppose that**
**** 


*C*(*x*) = 0.02*x*2 + 5*x* + 2,416

**is the total cost (in dollars) for a company to produce** ***x*** **units of a certain product.** 

Compute the

 **marginal and actual cost for** ***x*** **= 90.**
**** 
**Marginal cost is $** **and actual cost is $.**

To compute the **marginal cost** and the **actual cost** at $ x = 90 $, we'll proceed as follows:

### 1. **Marginal Cost**
The marginal cost is the derivative of the total cost function $ C(x) $ with respect to $ x $. The given cost function is:
\[
C(x) = 0.02x^2 + 5x + 2416
\]

First, let's differentiate $ C(x) $ with respect to $ x $:
\[
C'(x) = \frac{d}{dx}(0.02x^2 + 5x + 2416)
\]
\[
C'(x) = 2 \times 0.02x + 5 = 0.04x + 5
\]

Now, substitute $ x = 90 $ into the marginal cost formula:
\[
C'(90) = 0.04(90) + 5 = 3.6 + 5 = 8.6
\]
Thus, the marginal cost at $ x = 90 $ is **$8.60**.

### 2. **Actual Cost**
The actual cost is simply the total cost evaluated at $ x = 90 $. We substitute $ x = 90 $ into the original cost function $ C(x) $:
\[
C(90) = 0.02(90)^2 + 5(90) + 2416
\]
\[
C(90) = 0.02(8100) + 450 + 2416
\]
\[
C(90) = 162 + 450 + 2416 = 3028
\]
Thus, the actual cost at $ x = 90 $ is **$3028.00**.

### Final Answers:
- Marginal cost at $ x = 90 $ is **$8.60**.
- Actual cost at $ x = 90 $ is **$3028.00**.

Would you like more details or have any questions?  
Here are 5 related questions to explore further:
1. What is the interpretation of marginal cost in business economics?
2. How would the marginal cost change for $ x = 100 $?
3. What is the average cost at $ x = 90 $?
4. How would the actual cost change if the fixed cost component ($ 2416 $) were reduced?
5. How is the marginal cost used in pricing strategies?

**Tip:** The marginal cost represents the additional cost incurred for producing one more unit of the product.

Suppose that C(x) = 0.02x^2 + 5x + 2,416 is the total cost (in dollars) for a company to produce x units of a certain product. Compute the marginal and actual cost for x = 90.

Learn how to compute the marginal and actual cost for x = 90 units based on the cost function C(x) = 0.02x^2 + 5x + 2416. This problem involves calculus concepts like differentiation and cost functions, suitable for advanced high school students.

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