Assume that a competitive firm has the total cost function: TC=1q3−40q2+870q+1500 T C = 1 q 3 - 40 q 2 + 870 q + 1500 Suppose the price of the firm's output (sold in integer units) is $700 per unit. Use calculus and formulas to find a solution (don't just build a table in a spreadsheet as in the previous lesson). Hint 1: The first derivative of the total profit function, which is cumulative, is the marginal profit function, which is incremental. The lecture and formula summary explain how to compute the derivative. Set the marginal profit equal to zero to define an equation for the optimal quantity q. Rearrange the equation to the quadratic form aq2 + bq + c = 0, where a, b, and c represent numbers. Use the quadratic formula to solve for q: q=−b±b2−4ac‾‾‾‾‾‾‾‾√2a q = - b ± b 2 - 4 a c 2 a For non-integer quantity, round up and down to find the integer quantity with the optimal profit. Hint 2: When computing the total profit for each candidate quantity, use the total profit function you define (rather than summing the marginal profits using the marginal profit function).How many integer units should the firm produce to maximize profit? Please specify your answer as an integer. Answer (2 points) Save your answer What is the total profit at the optimal integer output level? Please specify your answer as an integer.

### Step 1: Define the Total Cost (TC) and Revenue (R) Functions

We are given the total cost (TC) function:

\[
TC(q) = 1q^3 - 40q^2 + 870q + 1500
\]

The revenue function is given by:

\[
R(q) = P \times q
\]

where $P = 700$ (the price per unit), so the revenue function becomes:

\[
R(q) = 700q
\]

### Step 2: Define the Total Profit Function

The total profit function $\pi(q)$ is the difference between revenue and total cost:

\[
\pi(q) = R(q) - TC(q)
\]

Substituting the expressions for $R(q)$ and $TC(q)$:

\[
\pi(q) = 700q - (q^3 - 40q^2 + 870q + 1500)
\]

Simplifying:

\[
\pi(q) = 700q - q^3 + 40q^2 - 870q - 1500
\]
\[
\pi(q) = -q^3 + 40q^2 - 170q - 1500
\]

### Step 3: Find the Marginal Profit Function

To find the quantity that maximizes profit, we take the first derivative of the total profit function (the marginal profit function) and set it equal to zero:

\[
\pi'(q) = \frac{d}{dq}(-q^3 + 40q^2 - 170q - 1500)
\]
\[
\pi'(q) = -3q^2 + 80q - 170
\]

### Step 4: Set the Marginal Profit Function Equal to Zero

Set the marginal profit equal to zero to find the critical points:

\[
-3q^2 + 80q - 170 = 0
\]

This is a quadratic equation of the form $aq^2 + bq + c = 0$, where $a = -3$, $b = 80$, and $c = -170$.

### Step 5: Solve the Quadratic Equation

We can use the quadratic formula:

\[
q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute $a = -3$, $b = 80$, and $c = -170$:

\[
q = \frac{-80 \pm \sqrt{80^2 - 4(-3)(-170)}}{2(-3)}
\]
\[
q = \frac{-80 \pm \sqrt{6400 - 2040}}{-6}
\]
\[
q = \frac{-80 \pm \sqrt{4360}}{-6}
\]
\[
q = \frac{-80 \pm 66.03}{-6}
\]

Now, solve for both roots:

1. $ q_1 = \frac{-80 + 66.03}{-6} = \frac{-13.97}{-6} = 2.33 $
2. $ q_2 = \frac{-80 - 66.03}{-6} = \frac{-146.03}{-6} = 24.34 $

### Step 6: Round to Integer Values

Since quantity must be an integer, we check $q = 2$ and $q = 24$.

### Step 7: Compute Total Profit for Each Integer Value

The total profit function is:

\[
\pi(q) = -q^3 + 40q^2 - 170q - 1500
\]

For $q = 2$:

\[
\pi(2) = -(2)^3 + 40(2)^2 - 170(2) - 1500
\]
\[
\pi(2) = -8 + 160 - 340 - 1500 = -1688
\]

For $q = 24$:

\[
\pi(24) = -(24)^3 + 40(24)^2 - 170(24) - 1500
\]
\[
\pi(24) = -13824 + 23040 - 4080 - 1500 = 3636
\]

### Step 8: Conclusion

The firm should produce **24 units** to maximize profit, and the total profit at this level is **$3636**.

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Do you have any questions or need further details?

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Here are 5 related questions to expand on this topic:

1. How does marginal cost relate to profit maximization?
2. What happens to profit if the price per unit changes?
3. How would the firm’s production decision change if fixed costs increased?
4. Can you explain the significance of the second derivative in profit maximization?
5. How does a change in variable cost impact the firm's output decision?

**Tip:** Always check both roots when solving quadratic equations to avoid missing potential solutions.

Assume that a competitive firm has the total cost function: TC=1q^3−40q^2+870q+1500. Suppose the price of the firm's output (sold in integer units) is $700 per unit. Use calculus and formulas to find a solution. How many integer units should the firm produce to maximize profit? What is the total profit at the optimal integer output level?

Explore how to maximize profit for a competitive firm given a total cost function TC = q^3 - 40q^2 + 870q + 1500 and price per unit at $700. This calculus-based solution includes solving the quadratic equation derived from the marginal profit function. Suitable for advanced high school and college students.

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