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Henry is starting a t-shirt printing business and plans on selling each shirt for $30, with his cost being $8 per shirt. The equipment will cost $1200. Henry orders 600 shirts and determines that the profit for his new business is modeled by the function P= 22x - 1200. What is the domain of this function in this context?

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To determine the domain of the profit function \( P = 22x - 1200 \) where \( x \) represents the number of shirts sold, we need to consider the context of the business.

  1. Understanding the parameters:

    • Each shirt is sold for $30.
    • The cost per shirt is $8, leading to a profit of $22 per shirt (since \( 30 - 8 = 22 \)).
    • Henry orders 600 shirts, which means he has an inventory of 600 shirts available to sell.

  2. Defining the variable \( x \):

    • The variable \( x \) represents the number of shirts sold. In this context, \( x \) can range from 0 (if no shirts are sold) to 600 (if all shirts are sold).

  3. Domain of the function:

    • Therefore, the domain of the function \( P \) (profit) will be the possible values of \( x \): \[ 0 \leq x \leq 600 \]

So, the domain of the profit function \( P = 22x - 1200 \) is \( [0, 600] \).