We are given the revenue equation $R = -750(x - 4)^2 + 192,000$, where $x$ represents the number of $5 price increases from the initial price of $60.

To solve this:

Step 1: Identify the vertex of the quadratic function

The equation is already in vertex form: $R = -750(x - 4)^2 + 192,000$ The vertex of a quadratic function in the form $R = a(x - h)^2 + k$ occurs at $x = h$, and the maximum value of $R$ occurs at that point if $a < 0$.

Here, $h = 4$, meaning that the maximum revenue occurs when $x = 4$.

Step 2: Calculate the price at maximum revenue

The price increases in steps of $5. Initially, the price was $60. Each increase corresponds to an increment of $5, so for $x = 4$:

$\text{New price} = 60 + 4 \times 5 = 60 + 20 = 80$

Thus, the shoe company should sell each pair of sneakers for $80 to earn the maximum revenue.

The correct answer is:

$\boxed{d. \, 80 \, \text{per pair of sneakers}}$

Would you like more details or have any other questions?

Here are 5 related questions to further your understanding:

1. How do we determine whether a quadratic equation represents a maximum or minimum?
2. What are the implications of changing the price on the number of units sold?
3. How can we interpret the vertex form of a quadratic equation in real-world problems?
4. What other types of models can be used to predict revenue based on price changes?
5. Why does increasing the price beyond a certain point decrease total revenue?

Tip: Quadratic functions always have a maximum or minimum, depending on the coefficient of the squared term, which is key in optimizing real-world problems!