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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:

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

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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!

A shoe company sells each pair of sneakers for $60 and expects to sell 3,000 pairs at this price. Their sales analysts have predicted the revenue, R, based on the number x, of $5 price increases using the equation R = -750(x - 4)^2 + 192,000. Assuming the sales analysts have predicted the revenue accurately, for what price should the shoe company sell each pair of sneakers in order to earn the maximum revenue?

Learn how to determine the price that maximizes revenue using quadratic functions. This problem involves the vertex form of a quadratic equation and illustrates a real-world application of revenue prediction based on price increases. Suitable for high school students in Grades 9-12.

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