Steven is starting a baseball hat printing business and plans on selling each hat for $20 with his cost being $8 per hat. The equipment will cost $800. Steven orders 500 hats and determines that the profit for his new business is modeled by the function P = 12x − 800.

Question

What is the domain of this function in this situation?
Responses
A x ≥ 20x ≥ 20
B all real numbersall real numbers
C {0 ≤ x ≤ 500}{0 ≤ x ≤ 500}
D {0 ≥ x ≥ 500}{0 ≥ x ≥ 500}
E {−800 ≤ x ≤ 5,200}

Answer 1

To find the appropriate domain for the profit function \( P = 12x - 800 \), we need to consider what \( x \) represents in this context. In this case, \( x \) represents the number of hats sold.

Given that Steven orders 500 hats, the maximum number of hats he can sell is 500. Additionally, he cannot sell a negative number of hats. Thus, the valid range for \( x \) is from 0 to 500.

This can be expressed as:

\[ {0 \leq x \leq 500} \]

Therefore, the correct response is:

C \({0 \leq x \leq 500}\)