Assume that the following has a linear cost function. 

Start 2 By 3 Table 1st Row 1st Column Fixed Cost 2nd Column Marginal Cost per item 3rd Column Item Sells For 2nd Row 1st Column $ 850 2nd Column $ 10 3rd Column $ 40 EndTable

Fixed Cost

Marginal Cost per item

Item Sells For

$850

$10

$40

Find the following.

​(a) the cost function

​(b) the revenue function

​(c) the profit function

​(d) the profit on 

99

items

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.

Assume that the following has a linear cost function. 

Fixed Cost: $850 
Marginal Cost per item: $10 
Item Sells For: $40 

Find the following: 
(a) the cost function 
(b) the revenue function 
(c) the profit function 
(d) the profit on 99 items.

Learn how to determine the cost, revenue, and profit functions for a product with a fixed cost of $850, marginal cost of $10 per item, and a selling price of $40. This guide includes step-by-step solutions and calculates the profit for selling 99 items. Suitable for high school students in Grades 9-12.

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