The latest demand equation for your "Banjos Rock" T-shirts is given by
q = −30x + 3600
where q is the number of shirts you can sell in one week if you charge x dollars per shirt. When you charge x dollars per shirt, your weekly cost function (in dollars) is given by
C(x) = −900x + 141750

To analyze the given functions, let's start by summarizing the equations:

1. **Demand Equation:**  
   $ q = -30x + 3600 $  
   This equation tells us the number of T-shirts ($ q $) sold in a week when the price per T-shirt is $ x $ dollars.

2. **Cost Function:**  
   $ C(x) = -900x + 141750 $  
   This equation represents the weekly cost in dollars when the price per T-shirt is $ x $ dollars.

Now, let's explore some key aspects:

### A. Revenue Function
The revenue function $ R(x) $ is given by multiplying the price per shirt ($ x $) by the number of shirts sold ($ q $):
\[
R(x) = x \cdot q = x(-30x + 3600) = -30x^2 + 3600x
\]

### B. Profit Function
The profit function $ P(x) $ is the difference between the revenue function and the cost function:
\[
P(x) = R(x) - C(x) = (-30x^2 + 3600x) - (-900x + 141750)
\]
Simplifying this, we get:
\[
P(x) = -30x^2 + 3600x + 900x - 141750 = -30x^2 + 4500x - 141750
\]

### C. Finding the Break-Even Points
To find the break-even points where profit is zero ($ P(x) = 0 $), solve:
\[
-30x^2 + 4500x - 141750 = 0
\]
Simplify by dividing everything by -30:
\[
x^2 - 150x + 4725 = 0
\]
Using the quadratic formula, $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $:
\[
x = \frac{150 \pm \sqrt{150^2 - 4 \cdot 1 \cdot 4725}}{2 \cdot 1} = \frac{150 \pm \sqrt{22500 - 18900}}{2} = \frac{150 \pm \sqrt{3600}}{2}
\]
\[
x = \frac{150 \pm 60}{2}
\]
So, the solutions are:
\[
x = \frac{150 + 60}{2} = 105 \quad \text{and} \quad x = \frac{150 - 60}{2} = 45
\]

Thus, the break-even points occur when the price per shirt is $45 or $105.

Would you like more details on these calculations or have any specific questions? Here are some related questions to consider:

1. What is the maximum profit, and at what price does it occur?
2. How many T-shirts are sold at the break-even points?
3. What happens to profit if the price is set below $45 or above $105?
4. How would changes in cost or demand equations affect the profit function?
5. What is the significance of the vertex of the profit parabola?

**Tip:** When analyzing profit functions, identifying the break-even points and the price for maximum profit can offer valuable insights for pricing strategies.

The latest demand equation for your 'Banjos Rock' T-shirts is given by q = −30x + 3600, where q is the number of shirts you can sell in one week if you charge x dollars per shirt. When you charge x dollars per shirt, your weekly cost function (in dollars) is given by C(x) = −900x + 141750.

Explore how to calculate break-even points and analyze profit for the demand equation q = -30x + 3600 and cost function C(x) = -900x + 141750. This problem involves algebraic concepts such as revenue, profit functions, and quadratic equations, perfect for students in Grades 10-12.

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