The mark department at Sports Stuff found that approximately 600 pairs of running shoes will be sold monthly when the average price of each pair of running shoes is $90. It was observed that for each $5 reduction in price, an additional 50 pairs of running shoes will be monthly. What price per pair of running shoes will maximize the store's monthly revenue from the sale of running shoes?

Answer

To find the price per pair of running shoes that maximizes the store's monthly revenue, we can follow these steps:

### Step 1: Define Variables
Let:
- \( p \) = price per pair of running shoes
- \( x \) = number of $5 reductions in price

### Step 2: Establish the Revenue Function
Initially, when the price is $90, the number of shoes sold is 600. For each $5 reduction in price, an additional 50 pairs are sold. Therefore, we can express the price and the number of shoes sold as:
- Price after \( x \) reductions: \( p = 90 - 5x \)
- Number of shoes sold after \( x \) reductions: \( 600 + 50x \)

The revenue \( R \) can be expressed as:
\[ R = \text{(Price per pair)} \times \text{(Number of pairs sold)} = (90 - 5x)(600 + 50x) \]

### Step 3: Expand the Revenue Function
Now, we will expand the revenue function:
\[ R = (90 - 5x)(600 + 50x) \]
Using the distributive property (FOIL method):
\[ R = 90 \cdot 600 + 90 \cdot 50x - 5x \cdot 600 - 5x \cdot 50x \]
Calculating each term:
\[ R = 54000 + 4500x - 3000x - 250x^2 \]
Combining like terms:
\[ R = -250x^2 + 1500x + 54000 \]

### Step 4: Find the Maximum Revenue
The revenue function \( R \) is a quadratic equation in the form \( R = Ax^2 + Bx + C \), where \( A = -250 \), \( B = 1500 \), and \( C = 54000 \). To find the maximum revenue, we use the vertex formula:
\[ x = -\frac{B}{2A} \]
Substituting the values of \( A \) and \( B \):
\[ x = -\frac{1500}{2 \cdot -250} = \frac{1500}{500} = 3 \]

### Step 5: Calculate the Optimal Price
Now that we have \( x = 3 \), we can find the price that maximizes revenue:
\[ p = 90 - 5x = 90 - 5 \cdot 3 = 90 - 15 = 75 \]

### Conclusion
The price per pair of running shoes that will maximize the store's monthly revenue is **$75**.

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