To solve this problem, we will follow these steps:

Step 1: Define the profit function

The profit function $P$ is given by: $P = R - C$ where $R$ is the revenue and $C$ is the cost.

Step 2: Find the revenue function

The revenue $R$ is the product of the price $p$ and the quantity $x$, which gives: $R = p \cdot x = (700 - \frac{1}{2}x) \cdot x = 700x - \frac{1}{2}x^2$

Step 3: Define the cost function

The cost function is given by: $C = 300 + 2x$

Step 4: Formulate the profit function

Substitute the expressions for $R$ and $C$ into the profit function: $P = (700x - \frac{1}{2}x^2) - (300 + 2x)$ Simplify: $P = 700x - \frac{1}{2}x^2 - 300 - 2x$ Thus, $P = -\frac{1}{2}x^2 + 698x - 300$

Step 5: Find the quantity that maximizes profit

To find the value of $x$ that maximizes profit, we need to take the derivative of $P$ with respect to $x$, set it equal to zero, and solve for $x$.

$\frac{dP}{dx} = -x + 698$ Set $\frac{dP}{dx} = 0$: $-x + 698 = 0$ Thus, $x = 698$

So, the quantity that maximizes profit is $698$ units.

Step 6: Find the selling price at this optimal quantity

Substitute $x = 698$ into the demand function: $p = 700 - \frac{1}{2}(698)$ $p = 700 - 349$ $p = 351$

Therefore, the selling price at this optimal quantity is $351 per unit.

Step 7: Find the maximum profit

Substitute $x = 698$ into the profit function: $P = -\frac{1}{2}(698)^2 + 698 \cdot 698 - 300$ $P = -\frac{1}{2}(487204) + 487204 - 300$ $P = -243602 + 487204 - 300$ $P = 243302$

Thus, the maximum profit is $243302.

Summary

(a) The quantity that will maximize profit is $698$ units.

(b) The selling price at this optimal quantity is $351 per unit.

(c) The maximum profit is $243302.

Would you like more details or have any questions?

Related Questions:

1. How would changes in the cost function affect the profit-maximizing quantity?
2. What happens to the profit if the demand function changes to a different form?
3. How does the elasticity of demand affect the revenue function in this scenario?
4. How would a fixed cost increase of $100 impact the maximum profit?
5. What would the profit function look like if there were a quadratic term in the cost equation?

Tip:

When maximizing profit, always find the derivative of the profit function and solve for the critical points to identify the optimal quantity.