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.