Frannie can sell 500 custom face masks each month at a price of $14.  For every $1 increase in price, she sells 10 fewer face masks.   

If x represents the number of $1 increases, and Revenue = Price * Quantity, then Frannie’s revenue can be calculated as:   R(x) = (14 + 1x)(500 – 10x) = 7000 + 360x – 10x2    

What price should she be charging for the masks to maximize her revenue?

To find the price Frannie should charge to maximize her revenue, we need to maximize the revenue function $ R(x) = 7000 + 360x - 10x^2 $, where $ x $ represents the number of $1 increases in price.

### Step 1: Recognize the quadratic form
The given revenue function is a quadratic equation in the form $ R(x) = -10x^2 + 360x + 7000 $. Since the coefficient of $ x^2 $ is negative (-10), the parabola opens downward, meaning the maximum revenue occurs at the vertex.

### Step 2: Use the vertex formula
For a quadratic equation in the form $ ax^2 + bx + c $, the x-coordinate of the vertex (which gives the value of $ x $ that maximizes or minimizes the function) is found using the formula:

\[
x = -\frac{b}{2a}
\]

Here, $ a = -10 $ and $ b = 360 $. Plugging these values into the formula:

\[
x = -\frac{360}{2(-10)} = \frac{360}{20} = 18
\]

So, Frannie should increase the price by $18 to maximize revenue.

### Step 3: Calculate the optimal price
The original price is $14, and since $ x = 18 $ represents an increase of $1 for each unit of $ x $:

\[
\text{Optimal price} = 14 + 18 = 32
\]

### Conclusion:
Frannie should charge **$32** per mask to maximize her revenue.

Would you like details on the calculation, or have any questions?

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Here are 5 related questions for further exploration:
1. What would Frannie's maximum revenue be at this price?
2. How many masks will Frannie sell at the optimal price?
3. What happens to the revenue if Frannie charges more than $32 per mask?
4. How would Frannie's revenue change if the original price was different (e.g., $16 instead of $14)?
5. How sensitive is the revenue to small changes in the price increase?

**Tip:** When solving optimization problems, the vertex formula is useful for finding maximum or minimum values in quadratic functions.

Frannie can sell 500 custom face masks each month at a price of $14. For every $1 increase in price, she sells 10 fewer face masks. If x represents the number of $1 increases, and Revenue = Price * Quantity, then Frannie’s revenue can be calculated as: R(x) = (14 + 1x)(500 – 10x) = 7000 + 360x – 10x^2. What price should she be charging for the masks to maximize her revenue?

Discover how to maximize revenue for Frannie's face mask business using quadratic functions. This detailed breakdown covers the revenue formula, solving for the vertex of a parabola, and calculating the optimal price to charge for maximum revenue. Suitable for Grades 9-12 algebra students.

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