Let's break this down step by step to find the weekly cost function $C(x)$, the weekly profit function $P(x)$, the price $x$ that maximizes profit, and the maximum profit.

Step 1: Find the weekly cost function $C(x)$

The total weekly cost includes the fixed cost of $400 (for using the facilities) and the cost of producing $q$ T-shirts, where each T-shirt costs $8.

The demand equation gives the number of T-shirts you sell as a function of the price: $q = -50x + 700$ So, the cost of producing $q$ T-shirts is: $8q = 8(-50x + 700) = -400x + 5600$

Now, the total weekly cost is the sum of the fixed cost ($400) and the production cost: $C(x) = 400 + (-400x + 5600) = -400x + 6000$ So, the cost function is: $C(x) = -400x + 6000$

Step 2: Find the weekly profit function $P(x)$

Profit is the difference between revenue and cost. First, let's find the revenue function.

The revenue from selling $q$ T-shirts at $x$ dollars each is: $R(x) = q \cdot x = (-50x + 700) \cdot x = -50x^2 + 700x$

Now, the profit function is: $P(x) = R(x) - C(x) = (-50x^2 + 700x) - (-400x + 6000)$ Simplifying: $P(x) = -50x^2 + 700x + 400x - 6000 = -50x^2 + 1100x - 6000$

So, the profit function is: $P(x) = -50x^2 + 1100x - 6000$

Step 3: Maximize the profit function

To find the price $x$ that maximizes the profit, we need to find the vertex of the quadratic profit function $P(x) = -50x^2 + 1100x - 6000$. The vertex occurs at: $x = \frac{-b}{2a}$ where $a = -50$ and $b = 1100$: $x = \frac{-1100}{2(-50)} = \frac{1100}{100} = 11$ So, the unit price that maximizes the profit is $x = 11$ dollars per T-shirt.

Step 4: Find the maximum profit

To find the maximum profit, substitute $x = 11$ into the profit function: $P(11) = -50(11)^2 + 1100(11) - 6000$ Calculating each term: $P(11) = -50(121) + 12100 - 6000 = -6050 + 12100 - 6000 = 50$ So, the maximum profit is $\boxed{50}$ dollars.

Summary:

• The cost function is: $C(x) = -400x + 6000$
• The profit function is: $P(x) = -50x^2 + 1100x - 6000$
• The price to charge for the maximum profit is: $x = 11$ dollars per T-shirt
• The maximum profit is: $50$ dollars

Do you have any questions or would you like more details?

Related Questions:

1. How would the cost function change if the production cost per T-shirt increased to $10?
2. What would happen to the profit function if the facility cost increased to $500 per week?
3. How does the demand equation affect the pricing strategy?
4. What is the break-even price where you neither make a profit nor incur a loss?
5. How would the maximum profit change if you sold 20 more shirts per week at the same price?

Tip:

Quadratic functions often model profit or revenue maximization problems, and the vertex formula is a key tool for finding maximum or minimum values efficiently.