Assume that a competitive firm has the total cost function:

TC=1q3−40q2+820q+1900
T
C
=
1
q
3
-
40
q
2
+
820
q
+
1900

Suppose the price of the firm's output (sold in integer units) is $650 per unit.

Create tables (but do not use calculus) with columns representing cost, revenue, and profit to find a solution.

How many units should the firm produce to maximize profit?

Please specify your answer as an integer.

Answer (2 points)

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What is the total profit at the optimal output level?

Please specify your answer as an integer.

To solve this problem, we'll calculate the total cost, total revenue, and total profit for different levels of output, $ q $, and identify the output level that maximizes profit.

### Given:
- Total Cost (TC) function: $ TC = q^3 - 40q^2 + 820q + 1900 $
- Price per unit (P): $650

### Formulas:
- **Total Revenue (TR)**: $ TR = P \times q $
- **Total Profit (π)**: $ \pi = TR - TC $

Let's calculate these values for $ q $ from 1 to 10 and find the maximum profit.

#### Step 1: Create the Table

| $ q $ | TC (Total Cost)        | TR (Total Revenue) | $ \pi $ (Profit)   |
|:------:|:---------------------:|:------------------:|:-------------------:|
|   1    | $ 1^3 - 40(1^2) + 820(1) + 1900 = 2681 $  | $ 650 \times 1 = 650 $    | $ 650 - 2681 = -2031 $ |
|   2    | $ 2^3 - 40(2^2) + 820(2) + 1900 = 3408 $  | $ 650 \times 2 = 1300 $   | $ 1300 - 3408 = -2108 $ |
|   3    | $ 3^3 - 40(3^2) + 820(3) + 1900 = 3977 $  | $ 650 \times 3 = 1950 $   | $ 1950 - 3977 = -2027 $ |
|   4    | $ 4^3 - 40(4^2) + 820(4) + 1900 = 4396 $  | $ 650 \times 4 = 2600 $   | $ 2600 - 4396 = -1796 $ |
|   5    | $ 5^3 - 40(5^2) + 820(5) + 1900 = 4673 $  | $ 650 \times 5 = 3250 $   | $ 3250 - 4673 = -1423 $ |
|   6    | $ 6^3 - 40(6^2) + 820(6) + 1900 = 4816 $  | $ 650 \times 6 = 3900 $   | $ 3900 - 4816 = -916 $  |
|   7    | $ 7^3 - 40(7^2) + 820(7) + 1900 = 4833 $  | $ 650 \times 7 = 4550 $   | $ 4550 - 4833 = -283 $  |
|   8    | $ 8^3 - 40(8^2) + 820(8) + 1900 = 4732 $  | $ 650 \times 8 = 5200 $   | $ 5200 - 4732 = 468 $   |
|   9    | $ 9^3 - 40(9^2) + 820(9) + 1900 = 4519 $  | $ 650 \times 9 = 5850 $   | $ 5850 - 4519 = 1331 $  |
|  10    | $ 10^3 - 40(10^2) + 820(10) + 1900 = 4202 $ | $ 650 \times 10 = 6500 $ | $ 6500 - 4202 = 2298 $  |

#### Step 2: Identify Maximum Profit

From the table, the maximum profit occurs at $ q = 10 $ with a total profit of **2298**.

### Answers:
1. **Optimal Output Level:** The firm should produce **10 units** to maximize profit.
2. **Total Profit at Optimal Output Level:** The total profit at this level is **2298**.

Would you like more details, or do you have any questions?

Here are 5 relative questions:
1. What happens to profit if the price per unit decreases?
2. How would fixed costs affect the optimal output level?
3. What if the cost function included a negative cubic term?
4. What is the impact on profit if the firm increases production beyond 10 units?
5. How would a change in variable costs impact the profit-maximizing output?

**Tip:** Always calculate total cost and revenue over a range of output levels to identify the one that maximizes profit.

Assume that a competitive firm has the total cost function: TC = q^3 − 40q^2 + 820q + 1900. Suppose the price of the firm's output (sold in integer units) is $650 per unit. Create tables (but do not use calculus) with columns representing cost, revenue, and profit to find a solution. How many units should the firm produce to maximize profit? What is the total profit at the optimal output level?

This problem explores how a competitive firm can maximize profit using cost and revenue analysis. The total cost function TC = q^3 - 40q^2 + 820q + 1900 is analyzed, and the optimal output level is determined by calculating total cost, total revenue, and total profit for integer values of q. The solution reveals that producing 10 units yields the maximum profit of $2298.

Math Problem Statement

TC=1q3−40q2+820q+1900 T C

1 q 3

Solution

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