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.