Assume that the following has a linear cost function. 
fixed cost:900$ Marginal cost per item: $12 item sells for: 30$ Find the following.
​(a) the cost function
​(b) the revenue function
​(c) the profit function
​(d) the profit on 102 items

Let's work through each part step by step.

### Given:
- **Fixed cost** = \$900
- **Marginal cost per item** = \$12
- **Selling price per item** = \$30

#### (a) The Cost Function

The total cost function $ C(x) $ is a sum of the fixed cost and the variable cost, where the variable cost is dependent on the number of items produced $ x $.

\[
C(x) = \text{Fixed cost} + (\text{Marginal cost} \times x)
\]
\[
C(x) = 900 + 12x
\]

So, the cost function is:

\[
C(x) = 900 + 12x
\]

#### (b) The Revenue Function

Revenue is the total amount of money generated from selling $ x $ items. The revenue function $ R(x) $ is simply the selling price per item multiplied by the number of items sold.

\[
R(x) = 30x
\]

#### (c) The Profit Function

Profit is the difference between revenue and cost. The profit function $ P(x) $ is:

\[
P(x) = R(x) - C(x)
\]
Substitute the expressions for $ R(x) $ and $ C(x) $:

\[
P(x) = 30x - (900 + 12x)
\]
Simplify the equation:

\[
P(x) = 30x - 900 - 12x
\]
\[
P(x) = 18x - 900
\]

So, the profit function is:

\[
P(x) = 18x - 900
\]

#### (d) The Profit on 102 Items

To find the profit on 102 items, substitute $ x = 102 $ into the profit function:

\[
P(102) = 18(102) - 900
\]
\[
P(102) = 1836 - 900
\]
\[
P(102) = 936
\]

So, the profit on 102 items is **\$936**.

### Summary of Functions:
- **Cost function**: $ C(x) = 900 + 12x $
- **Revenue function**: $ R(x) = 30x $
- **Profit function**: $ P(x) = 18x - 900 $
- **Profit on 102 items**: **\$936**

Would you like more details on any of these steps?

Here are some related questions to further explore:
1. How would the profit change if the marginal cost per item increased to \$15?
2. What is the break-even point (i.e., when profit is zero)?
3. How many items must be sold to achieve a profit of \$2000?
4. What is the revenue when 150 items are sold?
5. If the fixed cost decreases to \$800, how does that impact the profit function?

**Tip**: To maximize profits, you should analyze both the cost and revenue functions for large-scale production adjustments.

Assume that the following has a linear cost function. Fixed cost: $900, Marginal cost per item: $12, item sells for: $30. Find the following: (a) the cost function, (b) the revenue function, (c) the profit function, (d) the profit on 102 items.

Solve a linear cost function problem with a fixed cost of $900 and a marginal cost of $12 per item. Learn how to calculate the cost function, revenue function, profit function, and the profit on selling 102 items. Suitable for students in Grades 9-11.

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