The profit, P dollars, earned by students selling boxes of chocolates
is approximated to be P=-60x^2+600x , where x  dollars is the
selling price of one box of chocolates. What selling price gives the
maximum profit?

 a

$25

 b

$5

 c

$60

 d

$10

To determine the selling price that gives the maximum profit, we need to analyze the given quadratic function for profit:

\[
P(x) = -60x^2 + 600x
\]

where:
- $ P(x) $ represents the profit in dollars,
- $ x $ represents the selling price per box of chocolates in dollars.

### 1. Identify the Coefficients:
In the quadratic function $ P(x) = ax^2 + bx + c $, the coefficients are:
- $ a = -60 $
- $ b = 600 $
- $ c = 0 $ (since there is no constant term)

### 2. Find the Axis of Symmetry:
The selling price that maximizes profit occurs at the vertex of the parabola. The axis of symmetry for the parabola $ P(x) = ax^2 + bx + c $ is given by:

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

Substituting the values of $ a $ and $ b $:

\[
x = -\frac{600}{2(-60)} = \frac{600}{120} = 5
\]

### 3. Interpret the Result:
The selling price that gives the maximum profit is **$5**.

### 4. Answer:
Thus, the correct answer is:

**b) $5**

Would you like further details or have any questions?

---

### Further Exploration:
1. What would be the profit if the selling price is $10?
2. How much profit would be earned if the selling price is $5?
3. How does the profit change if the selling price increases to $7?
4. What happens to the profit if the selling price is below $5?
5. Can you determine the break-even points (selling prices where profit is zero)?
6. What effect does changing the coefficient of $ x^2 $ to -50 have on the maximum selling price?
7. How would the profit function look graphically? 
8. How is the shape of the parabola affected by the coefficients of the quadratic equation?

### Tip:
In profit maximization problems, the quadratic function often represents a parabola that opens downwards (indicated by a negative coefficient of $ x^2 $). The vertex of this parabola provides the maximum value of the function.

Learn how to maximize profit from selling chocolates using the quadratic function P(x) = -60x^2 + 600x. Find the selling price that gives the maximum profit with step-by-step instructions. Suitable for Grades 10-12.

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