The latest demand equation for your Yoda vs. Alien T-shirts is given by
q = −50x + 700
where q is the number of shirts you can sell in one week if you charge $x per shirt. The Student Council charges you $400 per week for use of their facilities, and the T-shirts cost you $8 each. Find the weekly cost as a function of the unit price x.
C(x) =

Hence, find the weekly profit as a function of x.
P(x) =

Determine the unit price (in dollars per T-shirt) you should charge to obtain the largest possible weekly profit.
x = $
per T-shirt
What is the largest possible weekly profit (in dollars)?
$

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.

The latest demand equation for your Yoda vs. Alien T-shirts is given by q = −50x + 700 where q is the number of shirts you can sell in one week if you charge $x per shirt. The Student Council charges you $400 per week for use of their facilities, and the T-shirts cost you $8 each. Find the weekly cost as a function of the unit price x. C(x) = Hence, find the weekly profit as a function of x. P(x) = Determine the unit price (in dollars per T-shirt) you should charge to obtain the largest possible weekly profit. x = $ per T-shirt What is the largest possible weekly profit (in dollars)?

Solve the problem of maximizing weekly profit for Yoda vs. Alien T-shirts with a given demand equation, cost, and revenue functions. Learn how to derive cost and profit functions, and determine the price per T-shirt for the largest possible profit using quadratic equations. Suitable for Grades 10-12.

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