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.