To solve this problem, we need to find the cost function, revenue function, profit function, and the profit for selling 99 items based on the provided data.

Given:

• Fixed Cost (F): $850
• Marginal Cost per item (M): $10
• Item Sells For (S): $40

(a) Cost Function

The cost function $C(x)$ represents the total cost of producing $x$ items. It's composed of the fixed cost and the variable cost (which is the marginal cost times the number of items produced).

$C(x) = F + M \cdot x$ Substitute the given values:

$C(x) = 850 + 10x$

(b) Revenue Function

The revenue function $R(x)$ represents the total income from selling $x$ items. This is simply the selling price per item times the number of items sold.

$R(x) = S \cdot x$ Substitute the given value:

$R(x) = 40x$

(c) Profit Function

The profit function $P(x)$ is found by subtracting the total cost from the total revenue.

$P(x) = R(x) - C(x)$ Substitute the functions from (a) and (b):

$P(x) = 40x - (850 + 10x)$ Simplify the expression:

$P(x) = 40x - 850 - 10x = 30x - 850$

(d) Profit on 99 Items

To find the profit when 99 items are sold, substitute $x = 99$ into the profit function.

$P(99) = 30(99) - 850$ Calculate:

$P(99) = 2970 - 850 = 2120$

Final Answers:

• (a) Cost Function: $C(x) = 850 + 10x$
• (b) Revenue Function: $R(x) = 40x$
• (c) Profit Function: $P(x) = 30x - 850$
• (d) Profit on 99 Items: $P(99) = 2120$

Would you like more details or have any questions?

Related Questions:

1. What is the break-even point (i.e., the number of items where profit is zero)?
2. How much profit would be made if 150 items were sold?
3. What would the cost be to produce 200 items?
4. How would the profit function change if the selling price per item increased to $50?
5. What is the marginal profit per item?

Tip:

Understanding the relationship between cost, revenue, and profit functions is crucial in making business decisions such as pricing and production levels.