Here is the step-by-step solution to the case-based question:

Given Data

1. Demand Function:

   $$q(p) = 500 - 20p$$

   where $p$ is the price per cup of coffee.

2. Revenue Function:

   $$R(p) = p \cdot q(p) = p(500 - 20p)$$

3. Cost Function:

   $$C(p) = 200 + 1.5 \cdot q(p) = 200 + 1.5(500 - 20p)$$
   

4. Profit Function:

   $$P(p) = R(p) - C(p)$$
   

1. Deriving the Revenue Function

$$R(p) = p \cdot q(p) = p(500 - 20p)$$

Simplify:

$$R(p) = 500p - 20p^2$$

So, the revenue function is:

$$R(p) = 500p - 20p^2$$

2. Deriving the Profit Function

First, find $C(p)<span\; style="color:\; var(--body-font-color);\; font-family:\; sans-serif;\; text-align:\; var(--bs-body-text-align);">:</span>$

$$C(p) = 200 + 1.5(500 - 20p)$$

Simplify:

$$C(p) = 200 + 750 - 30p$$
$$C(p) = 950 - 30p$$

Now, use $P(p) = R(p) - C(p)$:

$$P(p) = (500p - 20p^2) - (950 - 30p)$$

Simplify:

$$P(p) = 500p - 20p^2 - 950 + 30p$$
$$P(p) = -20p^2 + 530p - 950$$

So, the profit function is:

$$P(p) = -20p^2 + 530p - 950$$

3. Finding the Price That Maximizes Profit

To maximize profit, find the critical points of $P(p)$:

$$\frac{dP}{dp} = \frac{d}{dp}(-20p^2 + 530p - 950)$$
$$\frac{dP}{dp} = -40p + 530$$

Set $\frac{dP}{dp} = 0$

$$-40p + 530 = 0$$
$$p = \frac{530}{40} = 13.25$$

So, the price that maximizes profit is:

$$p = 13.25 \, \text{dollars}$$

4. Break-Even Price

At break-even, $P(p) = 0$:

$$-20p^2 + 530p - 950 = 0$$

Solve this quadratic equation using the quadratic formula:

$$p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Here, $a=-\mathrm{20, <span\; style="color:\; var(--body-font-color);\; font-family:\; sans-serif;\; text-align:\; var(--bs-body-text-align);">, </span>}$$b = 530$, and $c = -950$:

$$p = \frac{-530 \pm \sqrt{530^2 - 4(-20)(-950)}}{2(-20)}$$$$p = \frac{-530 \pm \sqrt{280900 - 76000}}{-40}$$$$p = \frac{-530 \pm \sqrt{204900}}{-40}$$$$p = \frac{-530 \pm 453.21}{-40}$$

Solve for two possible values of $p$:

1. $p = \frac{-530 + 453.21}{-40} = \frac{-76.79}{-40} = 1.92$
   
2. $p = \frac{-530 - 453.21}{-40} = \frac{-983.21}{-40} = 24.58$
   

So, the shop breaks even at:

$$p = 1.92 \, \text{dollars and } p = 24.58 \, \text{dollars.}$$

5. Critical Thinking Discussion

• Impact of Fixed Costs: If the fixed costs (e.g., rent) increase, the break-even prices will shift upward because the profit will decrease for the same price levels. This could make the lower break-even price infeasible.
• Impact of Variable Costs: If variable costs (e.g., coffee beans) increase, the optimal price to maximize profit will rise, as higher costs need to be offset by higher revenue.