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### Problem Statement Summary:
You are provided with a cost function $ C(x) $ for producing $ x $ units of a product. The cost function is:
\[
C(x) = 10x^2 + 80x + 500
\]
You need to find:
1. The total cost of producing 50 units.
2. The value of $ C(100) $.
3. The change in cost when production increases from 50 to 100 units.
4. The average cost per unit for producing 100 units.
5. The marginal cost at 100 units.

### Solutions:

1. **Total cost of producing 50 units:**
   \[
   C(50) = 10(50)^2 + 80(50) + 500
   \]
   \[
   C(50) = 10(2500) + 4000 + 500 = 25000 + 4000 + 500 = 29500
   \]
   The total cost of producing 50 units is \$29,500.

2. **Value of $ C(100) $:**
   \[
   C(100) = 10(100)^2 + 80(100) + 500
   \]
   \[
   C(100) = 10(10000) + 8000 + 500 = 100000 + 8000 + 500 = 108500
   \]
   The value of $ C(100) $ is \$108,500.

3. **Change in cost from 50 to 100 units:**
   \[
   \text{Change in cost} = C(100) - C(50) = 108500 - 29500 = 79000
   \]
   The change in cost when production increases from 50 to 100 units is \$79,000.

4. **Average cost per unit for 100 units:**
   \[
   \text{Average cost} = \frac{C(100)}{100} = \frac{108500}{100} = 1085
   \]
   The average cost per unit for producing 100 units is \$1,085.

5. **Marginal cost at 100 units:**
   The marginal cost is given by the derivative of the cost function:
   \[
   C'(x) = \frac{d}{dx}[10x^2 + 80x + 500] = 20x + 80
   \]
   At $ x = 100 $:
   \[
   C'(100) = 20(100) + 80 = 2000 + 80 = 2080
   \]
   The marginal cost at 100 units is \$2,080.

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### Relative Questions:
1. How would the total cost change if the production increases to 150 units?
2. What is the average cost per unit when producing 50 units?
3. How does the marginal cost behave as production increases?
4. What is the fixed cost in this cost function?
5. How would the cost function change if the variable cost per unit increases?

### Tip:
Remember, the marginal cost is crucial in deciding the optimal level of production. It helps determine whether producing additional units will be profitable.

You are provided with a cost function C(x) for producing x units of a product. The cost function is C(x) = 10x^2 + 80x + 500. You need to find: 1. The total cost of producing 50 units. 2. The value of C(100). 3. The change in cost when production increases from 50 to 100 units. 4. The average cost per unit for producing 100 units. 5. The marginal cost at 100 units.

Explore how to solve a cost function problem using a quadratic equation. Learn how to calculate the total cost of producing 50 units, the value of C(100), the change in cost when production increases, the average cost per unit, and the marginal cost at 100 units. This guide is ideal for students in Grades 10-12.

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